Daily Papers

Robustness to adversarial attack is typically evaluated with adversarial accuracy. This metric quantifies the number of points for which, given a threat model, successful adversarial perturbations cannot be found. While essential, this metric does not capture all aspects of robustness and in particular leaves out the question of how many perturbations can be found for each point. In this work we introduce an alternative approach, adversarial sparsity, which quantifies how difficult it is to find a successful perturbation given both an input point and a constraint on the direction of the perturbation. This constraint may be angular (L2 perturbations), or based on the number of pixels (Linf perturbations). We show that sparsity provides valuable insight on neural networks in multiple ways. analyzing the sparsity of existing robust models illustrates important differences between them that accuracy analysis does not, and suggests approaches for improving their robustness. When applying broken defenses effective against weak attacks but not strong ones, sparsity can discriminate between the totally ineffective and the partially effective defenses. Finally, with sparsity we can measure increases in robustness that do not affect accuracy: we show for example that data augmentation can by itself increase adversarial robustness, without using adversarial training.

Building Virtual Cells that can accurately simulate cellular responses to perturbations is a long-standing goal in systems biology. A fundamental challenge is that high-throughput single-cell sequencing is destructive: the same cell cannot be observed both before and after a perturbation. Thus, perturbation prediction requires mapping unpaired control and perturbed populations. Existing models address this by learning maps between distributions, but typically assume a single fixed response distribution when conditioned on observed cellular context (e.g., cell type) and the perturbation type. In reality, responses vary systematically due to unobservable latent factors such as microenvironmental fluctuations and complex batch effects, forming a manifold of possible distributions for the same observed conditions. To account for this variability, we introduce PerturbDiff, which shifts modeling from individual cells to entire distributions. By embedding distributions as points in a Hilbert space, we define a diffusion-based generative process operating directly over probability distributions. This allows PerturbDiff to capture population-level response shifts across hidden factors. Benchmarks on established datasets show that PerturbDiff achieves state-of-the-art performance in single-cell response prediction and generalizes substantially better to unseen perturbations. See our project page (https://katarinayuan.github.io/PerturbDiff-ProjectPage/), where code and data will be made publicly available (https://github.com/DeepGraphLearning/PerturbDiff).

The problem of attribution is concerned with identifying the parts of an input that are responsible for a model's output. An important family of attribution methods is based on measuring the effect of perturbations applied to the input. In this paper, we discuss some of the shortcomings of existing approaches to perturbation analysis and address them by introducing the concept of extremal perturbations, which are theoretically grounded and interpretable. We also introduce a number of technical innovations to compute extremal perturbations, including a new area constraint and a parametric family of smooth perturbations, which allow us to remove all tunable hyper-parameters from the optimization problem. We analyze the effect of perturbations as a function of their area, demonstrating excellent sensitivity to the spatial properties of the deep neural network under stimulation. We also extend perturbation analysis to the intermediate layers of a network. This application allows us to identify the salient channels necessary for classification, which, when visualized using feature inversion, can be used to elucidate model behavior. Lastly, we introduce TorchRay, an interpretability library built on PyTorch.

Deep Learning using the eponymous deep neural networks (DNNs) has become an attractive approach towards various data-based problems of theoretical physics in the past decade. There has been a clear trend to deeper architectures containing increasingly more powerful and involved layers. Contrarily, Taylor coefficients of DNNs still appear mainly in the light of interpretability studies, where they are computed at most to first order. However, especially in theoretical physics numerous problems benefit from accessing higher orders, as well. This gap motivates a general formulation of neural network (NN) Taylor expansions. Restricting our analysis to multilayer perceptrons (MLPs) and introducing quantities we refer to as propagators and vertices, both depending on the MLP's weights and biases, we establish a graph-theoretical approach. Similarly to Feynman rules in quantum field theories, we can systematically assign diagrams containing propagators and vertices to the corresponding partial derivative. Examining this approach for S-wave scattering lengths of shallow potentials, we observe NNs to adapt their derivatives mainly to the leading order of the target function's Taylor expansion. To circumvent this problem, we propose an iterative NN perturbation theory. During each iteration we eliminate the leading order, such that the next-to-leading order can be faithfully learned during the subsequent iteration. After performing two iterations, we find that the first- and second-order Born terms are correctly adapted during the respective iterations. Finally, we combine both results to find a proxy that acts as a machine-learned second-order Born approximation.

This project focuses on a method to extract a frequency comb in mechanical means, for general interest and numerous practical applications in MEMS. The method of execution is the implementation of a beam that is exhibiting non-linear dynamics that is perturbed and analyzed for its transverse vibrations. The perturbation is an external harmonic driver with a chosen small amplitude and frequency (which is slightly detuned from the beam eigenfrequency), that when engaged with the unperturbed beam oscillations, causes it reach a state of "injection pulling" - an effect that occurs when one harmonic oscillator is coupled with a second one and causes it to oscillate in a frequency near its own. This causes the beam to reach SNIC bifurcation, rendering a frequency comb as desired. Theoretical analysis showed that the problem can be modelled using a non-linear equation of the beam, that translates to a form of the non-linear Duffing equation. While a solution to the dynamics function of the beam is hard to obtain in practice due to mathematical difficulties, a slow evolution model is suggested that is composed of functions of a amplitude and phase. Using several additional mathematical assumptions, the amplitude is seen to be related to the phase, while the phase equation solution is seen to be of the form of Adler's equation. These assumptions ultimately reduce the entire behaviour of the beam to a relatively simple solution to the Adler equation, which has a known analytical solution. Computerized numerical simulations are run on it to check the results and compare them to the theory and desired outcome. The results agreed with the theory and produce the expected frequency comb, showing the assumptions to be valid in extracting the comb.

Large language models (LLMs) have transformed natural language processing, but their reliable deployment requires effective uncertainty quantification (UQ). Existing UQ methods are often heuristic and lack a probabilistic foundation. This paper begins by providing a theoretical justification for the role of perturbations in UQ for LLMs. We then introduce a dual random walk perspective, modeling input-output pairs as two Markov chains with transition probabilities defined by semantic similarity. Building on this, we propose a fully probabilistic framework based on an inverse model, which quantifies uncertainty by evaluating the diversity of the input space conditioned on a given output through systematic perturbations. Within this framework, we define a new uncertainty measure, Inv-Entropy. A key strength of our framework is its flexibility: it supports various definitions of uncertainty measures, embeddings, perturbation strategies, and similarity metrics. We also propose GAAP, a perturbation algorithm based on genetic algorithms, which enhances the diversity of sampled inputs. In addition, we introduce a new evaluation metric, Temperature Sensitivity of Uncertainty (TSU), which directly assesses uncertainty without relying on correctness as a proxy. Extensive experiments demonstrate that Inv-Entropy outperforms existing semantic UQ methods. The code to reproduce the results can be found at https://github.com/UMDataScienceLab/Uncertainty-Quantification-for-LLMs.

Linear relaxation based perturbation analysis (LiRPA) for neural networks, which computes provable linear bounds of output neurons given a certain amount of input perturbation, has become a core component in robustness verification and certified defense. The majority of LiRPA-based methods focus on simple feed-forward networks and need particular manual derivations and implementations when extended to other architectures. In this paper, we develop an automatic framework to enable perturbation analysis on any neural network structures, by generalizing existing LiRPA algorithms such as CROWN to operate on general computational graphs. The flexibility, differentiability and ease of use of our framework allow us to obtain state-of-the-art results on LiRPA based certified defense on fairly complicated networks like DenseNet, ResNeXt and Transformer that are not supported by prior works. Our framework also enables loss fusion, a technique that significantly reduces the computational complexity of LiRPA for certified defense. For the first time, we demonstrate LiRPA based certified defense on Tiny ImageNet and Downscaled ImageNet where previous approaches cannot scale to due to the relatively large number of classes. Our work also yields an open-source library for the community to apply LiRPA to areas beyond certified defense without much LiRPA expertise, e.g., we create a neural network with a probably flat optimization landscape by applying LiRPA to network parameters. Our opensource library is available at https://github.com/KaidiXu/auto_LiRPA.

A central goal in systems biology and drug discovery is to predict the transcriptional response of cells to perturbations. This task is challenging due to the noisy and sparse nature of single-cell measurements, as well as the fact that perturbations often induce population-level shifts rather than changes in individual cells. Existing deep learning methods typically assume cell-level correspondences, limiting their ability to capture such global effects. We present scDFM, a generative framework based on conditional flow matching that models the full distribution of perturbed cells conditioned on control states. By incorporating a maximum mean discrepancy (MMD) objective, our method aligns perturbed and control populations beyond cell-level correspondences. To further improve robustness to sparsity and noise, we introduce the Perturbation-Aware Differential Transformer (PAD-Transformer), a backbone architecture that leverages gene interaction graphs and differential attention to capture context-specific expression changes. Across multiple genetic and drug perturbation benchmarks, scDFM consistently outperforms prior methods, demonstrating strong generalization in both unseen and combinatorial settings. In the combinatorial setting, it reduces mean squared error by 19.6% relative to the strongest baseline. These results highlight the importance of distribution-level generative modeling for robust in silico perturbation prediction. The code is available at https://github.com/AI4Science-WestlakeU/scDFM

Existing works have extensively studied adversarial examples, which are minimal perturbations that can mislead the output of deep neural networks (DNNs) while remaining imperceptible to humans. However, in this work, we reveal the existence of a harmless perturbation space, in which perturbations drawn from this space, regardless of their magnitudes, leave the network output unchanged when applied to inputs. Essentially, the harmless perturbation space emerges from the usage of non-injective functions (linear or non-linear layers) within DNNs, enabling multiple distinct inputs to be mapped to the same output. For linear layers with input dimensions exceeding output dimensions, any linear combination of the orthogonal bases of the nullspace of the parameter consistently yields no change in their output. For non-linear layers, the harmless perturbation space may expand, depending on the properties of the layers and input samples. Inspired by this property of DNNs, we solve for a family of general perturbation spaces that are redundant for the DNN's decision, and can be used to hide sensitive data and serve as a means of model identification. Our work highlights the distinctive robustness of DNNs (i.e., consistency under large magnitude perturbations) in contrast to adversarial examples (vulnerability for small imperceptible noises).

Training deep neural networks for classification often includes minimizing the training loss beyond the zero training error point. In this phase of training, a "neural collapse" behavior has been observed: the variability of features (outputs of the penultimate layer) of within-class samples decreases and the mean features of different classes approach a certain tight frame structure. Recent works analyze this behavior via idealized unconstrained features models where all the minimizers exhibit exact collapse. However, with practical networks and datasets, the features typically do not reach exact collapse, e.g., because deep layers cannot arbitrarily modify intermediate features that are far from being collapsed. In this paper, we propose a richer model that can capture this phenomenon by forcing the features to stay in the vicinity of a predefined features matrix (e.g., intermediate features). We explore the model in the small vicinity case via perturbation analysis and establish results that cannot be obtained by the previously studied models. For example, we prove reduction in the within-class variability of the optimized features compared to the predefined input features (via analyzing gradient flow on the "central-path" with minimal assumptions), analyze the minimizers in the near-collapse regime, and provide insights on the effect of regularization hyperparameters on the closeness to collapse. We support our theory with experiments in practical deep learning settings.

There has been a recent surge of interest in modeling neural networks (NNs) as Gaussian processes. In the limit of a NN of infinite width the NN becomes equivalent to a Gaussian process. Here we demonstrate that for an ensemble of large, finite, fully connected networks with a single hidden layer the distribution of outputs at initialization is well described by a Gaussian perturbed by the fourth Hermite polynomial for weights drawn from a symmetric distribution. We show that the scale of the perturbation is inversely proportional to the number of units in the NN and that higher order terms decay more rapidly, thereby recovering the Edgeworth expansion. We conclude by observing that understanding how this perturbation changes under training would reveal the regimes in which the Gaussian process framework is valid to model NN behavior.

Instruction-tuning plays a vital role in enhancing the task-solving abilities of large language models (LLMs), improving their usability in generating helpful responses on various tasks. However, previous work has demonstrated that they are sensitive to minor variations in instruction phrasing. In this paper, we explore whether introducing perturbations in instruction-tuning data can enhance LLMs' resistance against noisy instructions. We focus on how instruction-tuning with perturbations, such as removing stop words or shuffling words, affects LLMs' performance on the original and perturbed versions of widely-used benchmarks (MMLU, BBH, GSM8K). We further assess learning dynamics and potential shifts in model behavior. Surprisingly, our results suggest that instruction-tuning on perturbed instructions can, in some cases, improve downstream performance. These findings highlight the importance of including perturbed instructions in instruction-tuning, which can make LLMs more resilient to noisy user inputs.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

Predicting how genetic perturbations change cellular state is a core problem for building controllable models of gene regulation. Perturbations targeting the same gene can produce different transcriptional responses depending on their genomic locus, including different transcription start sites and regulatory elements. Gene-level perturbation models collapse these distinct interventions into the same representation. We introduce STRAND, a generative model that predicts single-cell transcriptional responses by conditioning on regulatory DNA sequence. STRAND represents a perturbation by encoding the sequence at its genomic locus and uses this representation to parameterize a conditional transport process from control to perturbed cell states. Representing perturbations by sequence, rather than by a fixed set of gene identifiers, supports zero-shot inference at loci not seen during training and expands inference-time genomic coverage from ~1.5% for gene-level single-cell foundation models to ~95% of the genome. We evaluate STRAND on CRISPR perturbation datasets in K562, Jurkat, and RPE1 cells. STRAND improves discrimination scores by up to 33% in low-sample regimes, achieves the best average rank on unseen gene perturbation benchmarks, and improves transfer to novel cell lines by up to 0.14 in Pearson correlation. Ablations isolate the gains to sequence conditioning and transport, and case studies show that STRAND resolves functionally alternative transcription start sites missed by gene-level models.

Harmful fine-tuning issue qi2023fine poses serious safety concerns for Large language models' fine-tuning-as-a-service. While existing defenses huang2024vaccine,rosati2024representation have been proposed to mitigate the issue, their performances are still far away from satisfactory, and the root cause of the problem has not been fully recovered. For the first time in the literature, we in this paper show that harmful perturbation over the model weights should be the root cause of alignment-broken of harmful fine-tuning. In order to attenuate the negative impact of harmful perturbation, we propose an alignment-stage solution, dubbed Booster. Technically, along with the original alignment loss, we append a loss regularizer in the alignment stage's optimization. The regularizer ensures that the model's harmful loss reduction before/after simulated harmful perturbation is attenuated, thereby mitigating the subsequent fine-tuning risk. Empirical results show that Booster can effectively reduce the harmful score of the fine-tuned models while maintaining the performance of downstream tasks. Our code is available at https://github.com/git-disl/Booster.

Accurate prediction of extreme weather events remains a major challenge for artificial intelligence-based weather prediction systems. While deterministic models such as FuXi, GraphCast, and SFNO have achieved competitive forecast skill relative to numerical weather prediction, their ability to represent uncertainty and capture extremes is still limited. This study investigates how state-of-the-art deterministic artificial intelligence-based models respond to initial-condition perturbations and evaluates the resulting ensembles in forecasting extremes. Using four perturbation strategies (Gaussian, Perlin noise, Hemispheric Centered Bred Vectors, and Huge Ensembles), we generate 50 member ensembles for the August 2022 Pakistan floods and China heatwave, and complement these case studies with a global threshold-based evaluation. Ensemble skill is assessed against ERA5 and compared with IFS ENS and the AIFS ENS probabilistic model using deterministic and probabilistic metrics. Results show that simpler perturbations like Gaussian and Perlin noise produce similarly realistic ensemble spread and probabilistic skill as flow-based approaches like HCBV and HENS, narrowing but not closing the performance gap with numerical weather prediction ensembles, or native probabilistic models which retain the highest probabilistic skill across variables. Model choice is the dominant factor for ensemble performance, not perturbation method. Across variables, models capture temperature extremes more effectively than precipitation. These findings demonstrate that simple input perturbations can extend deterministic models toward probabilistic forecasting in hardware-constrained settings, supporting artificial intelligence-driven early warning systems.

Predicting high-dimensional transcriptional responses to genetic perturbations is challenging due to severe experimental noise and sparse gene-level effects. Existing methods often suffer from mean collapse, where high correlation is achieved by predicting global average expression rather than perturbation-specific responses, leading to many false positives and limited biological interpretability. Recent approaches incorporate biological knowledge graphs into perturbation models, but these graphs are typically treated as dense and static, which can propagate noise and obscure true perturbation signals. We propose AdaPert, a perturbation-conditioned framework that addresses mean collapse by explicitly modeling sparsity and biological structure. AdaPert learns perturbation-specific subgraphs from biological knowledge graphs and applies adaptive learning to separate true signals from noise. Across multiple genetic perturbation benchmarks, AdaPert consistently outperforms existing baselines and achieves substantial improvements on DEG-aware evaluation metrics, indicating more accurate recovery of perturbation-specific transcriptional changes.

It is not fully understood why adversarial examples can deceive neural networks and transfer between different networks. To elucidate this, several studies have hypothesized that adversarial perturbations, while appearing as noises, contain class features. This is supported by empirical evidence showing that networks trained on mislabeled adversarial examples can still generalize well to correctly labeled test samples. However, a theoretical understanding of how perturbations include class features and contribute to generalization is limited. In this study, we provide a theoretical framework for understanding learning from perturbations using a one-hidden-layer network trained on mutually orthogonal samples. Our results highlight that various adversarial perturbations, even perturbations of a few pixels, contain sufficient class features for generalization. Moreover, we reveal that the decision boundary when learning from perturbations matches that from standard samples except for specific regions under mild conditions. The code is available at https://github.com/s-kumano/learning-from-adversarial-perturbations.

Despite the rapid progress of multimodal large language models (MLLMs), they have largely overlooked the importance of visual processing. In a simple yet revealing experiment, we interestingly find that language-only models, when provided with image captions, can achieve comparable or even better performance than MLLMs that consume raw visual inputs. This suggests that current MLLMs may generate accurate visual descriptions but fail to effectively integrate them during reasoning. Motivated by this, we propose a simple visual perturbation framework that enhances perceptual robustness without requiring algorithmic modifications or additional training data. Our approach introduces three targeted perturbations: distractor concatenation, dominance-preserving mixup, and random rotation, that can be easily integrated into existing post-training pipelines including SFT, DPO, and GRPO. Through extensive experiments across multiple datasets, we demonstrate consistent improvements in mathematical reasoning performance, with gains comparable to those achieved through algorithmic changes. Additionally, we achieve competitive performance among open-source 7B RL-tuned models by training Qwen2.5-VL-7B with visual perturbation. Through comprehensive ablation studies, we analyze the effectiveness of different perturbation strategies, revealing that each perturbation type contributes uniquely to different aspects of visual reasoning. Our findings highlight the critical role of visual perturbation in multimodal mathematical reasoning: better reasoning begins with better seeing. Our code is available at https://github.com/YutingLi0606/Vision-Matters.

We examine the geometry of neural network training using the Jacobian of trained network parameters with respect to their initial values. Our analysis reveals low-dimensional structure in the training process which is dependent on the input data but largely independent of the labels. We find that the singular value spectrum of the Jacobian matrix consists of three distinctive regions: a "chaotic" region of values orders of magnitude greater than one, a large "bulk" region of values extremely close to one, and a "stable" region of values less than one. Along each bulk direction, the left and right singular vectors are nearly identical, indicating that perturbations to the initialization are carried through training almost unchanged. These perturbations have virtually no effect on the network's output in-distribution, yet do have an effect far out-of-distribution. While the Jacobian applies only locally around a single initialization, we find substantial overlap in bulk subspaces for different random seeds. Our code is available at https://github.com/EleutherAI/training-jacobian

Large linear systems play an important role in high-energy theory, appearing in amplitude bootstraps and during integral reduction. This paper introduces FiniteFieldSolve, a general-purpose toolkit for exactly solving large linear systems over the rationals. The solver interfaces directly with Mathematica, is straightforward to install, and seamlessly replaces Mathematica's native solvers. In testing, FiniteFieldSolve is approximately two orders of magnitude faster than Mathematica and uses an order of magnitude less memory. The package also compares favorably against other public solvers in FiniteFieldSolve's intended use cases. As the name of the package suggests, solutions are obtained via well-known finite field methods. These methods suffer from introducing an inordinate number of modulo (or integer division) operations with respect to different primes. By automatically recompiling itself for each prime, FiniteFieldSolve converts the division operations into much faster combinations of instructions, dramatically improving performance. The technique of compiling the prime can be applied to any finite field solver, where the time savings will be solver dependent. The operation of the package is illustrated through a detailed example of an amplitude bootstrap.

Flow-based generative models have been employed for sampling the Boltzmann distribution, but their application to high-dimensional systems is hindered by the significant computational cost of obtaining the Jacobian of the flow. To overcome this challenge, we introduce the flow perturbation method, which incorporates optimized stochastic perturbations into the flow. By reweighting trajectories generated by the perturbed flow, our method achieves unbiased sampling of the Boltzmann distribution with orders of magnitude speedup compared to both brute force Jacobian calculations and the Hutchinson estimator. Notably, it accurately sampled the Chignolin protein with all atomic Cartesian coordinates explicitly represented, which, to our best knowledge, is the largest molecule ever Boltzmann sampled in such detail using generative models.

Perturbation-based post-hoc image explanation methods are commonly used to explain image prediction models. These methods perturb parts of the input to measure how those parts affect the output. Since the methods only require the input and output, they can be applied to any model, making them a popular choice to explain black-box models. While many different methods exist and have been compared with one another, it remains poorly understood which parameters of the different methods are responsible for their varying performance. This work uses the Randomized Input Sampling for Explanations (RISE) method as a baseline to evaluate many combinations of mask sampling, segmentation techniques, smoothing, attribution calculation, and per-segment or per-pixel attribution, using a proxy metric. The results show that attribution calculation, which is frequently the focus of other works, has little impact on the results. Conversely, segmentation and per-pixel attribution, rarely examined parameters, have a significant impact. The implementation of and data gathered in this work are available online: https://github.com/guspih/post-hoc-image-perturbation and https://bit.ly/smooth-mask-perturbation.

Recent benchmarks reveal that models for single-cell perturbation response are often outperformed by simply predicting the dataset mean. We trace this anomaly to a metric artifact: control-referenced deltas and unweighted error metrics reward mode collapse whenever the control is biased or the biological signal is sparse. Large-scale in silico simulations and analysis of two real-world perturbation datasets confirm that shared reference shifts, not genuine biological change, drives high performance in these evaluations. We introduce differentially expressed gene (DEG)-aware metrics, weighted mean-squared error (WMSE) and weighted delta R^{2} (R^{2}_{w}(Δ)) with respect to all perturbations, that measure error in niche signals with high sensitivity. We further introduce negative and positive performance baselines to calibrate these metrics. With these improvements, the mean baseline sinks to null performance while genuine predictors are correctly rewarded. Finally, we show that using WMSE as a loss function reduces mode collapse and improves model performance.

We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the Wronskian of suitable solutions of the two underlying difference equations. Until now only the case of perturbations of the main diagonal was known. We extend the known results to arbitrary perturbations, allow any (half-)open and closed spectral intervals, simplify the proof, and establish the comparison theorem.

Existing evaluation methods largely rely on clean, static benchmarks, which can overestimate true model performance by failing to capture the noise and variability inherent in real-world user inputs. This is especially true for language models, which can face human-generated text queries containing mistakes, typos, or alternative ways of phrasing the same question. In this work, we introduce a theoretical framework for quantifying model sensitivity to prompt variants, or brittleness, that can enable us to disentangle data-induced difficulty from prompt-related variability. Using this framework, we design a novel evaluation pipeline, Brittlebench, to holistically evaluate the sensitivity of frontier models. We apply semantics-preserving perturbations to a suite of popular benchmarks, and observe model performance to degrade as much as 12%. However, these perturbations do not affect all models equally: even a single perturbation alters the relative ranking of models in 63% of cases, impacting conclusions about comparative model performance. Decomposing the total variance of both state-of-the-art open-weight and commercial models, we find that semantics-preserving input perturbations can account for up to half of the performance variance for a given model. Brittlebench highlights the need for more robust evaluations and models, and allows us to systematically understand model brittleness.

While Large Language Models (LLMs) achieve high performance on standard mathematical benchmarks, their underlying reasoning processes remain highly overfit to standard textual formatting. We propose a perturbation pipeline consisting of 14 techniques to evaluate robustness of LLM reasoning. We apply this pipeline to AIME 2024 dataset and evalute 8 state-of-the-art models on the resulting benchmark. While frontier models exhibit resilience, open weights reasoning models suffer catastrophic collapses (up to 55% average accuracy drops across perturbations and up to 100% on some), exposing structural fragility. To further disentangle mechanical parsing failures from downstream reasoning failures, we strictly isolate the models' working memory capacity by forcing models to solve multiple unperturbed mathematical problems sequentially within a single context window. Our results indicate that open weight models ranging from 7B to 120B parameters and Claude Opus 4.6 exhibit accuracy decay on subsequent problems. This degradation demonstrates that intermediate reasoning steps permanently pollute standard dense attention mechanisms. We argue that to achieve reliable reasoning, future reasoning architectures must integrate explicit contextual resets within a model's own Chain-of-Thought, leading to fundamental open questions regarding the optimal granularity of atomic reasoning tasks.

Large Language Models (LLMs) heavily rely on high-quality training data, making data valuation crucial for optimizing model performance, especially when working within a limited budget. In this work, we aim to offer a third-party data valuation approach that benefits both data providers and model developers. We introduce a linearized future influence kernel (LinFiK), which assesses the value of individual data samples in improving LLM performance during training. We further propose ALinFiK, a learning strategy to approximate LinFiK, enabling scalable data valuation. Our comprehensive evaluations demonstrate that this approach surpasses existing baselines in effectiveness and efficiency, demonstrating significant scalability advantages as LLM parameters increase.

We analyze the DP (differential privacy) properties of the quantum recommendation algorithm and the quantum-inspired-classical recommendation algorithm. We discover that the quantum recommendation algorithm is a privacy curating mechanism on its own, requiring no external noise, which is different from traditional differential privacy mechanisms. In our analysis, a novel perturbation method tailored for SVD (singular value decomposition) and low-rank matrix approximation problems is introduced. Using the perturbation method and random matrix theory, we are able to derive that both the quantum and quantum-inspired-classical algorithms are big(mathcal{O}big(frac 1nbig),,, mathcal{O}big(1{min{m,n}}big)big)-DP under some reasonable restrictions, where m and n are numbers of users and products in the input preference database respectively. Nevertheless, a comparison shows that the quantum algorithm has better privacy preserving potential than the classical one.

Single-cell perturbation models can reduce costly wet-lab screening by predicting how cells respond transcriptionally to interventions. While recent generative models improve population-level prediction, individual generated cells are not explicitly checked for biological consistency. We introduce PerturbCellRL, a reinforcement learning (RL) framework that post-trains a pretrained single-cell transcriptomic generator using a suite of cell-level verifiers as rewards. These verifiers define four rewards: Pearson top-k similarity, RMSE top-k proximity, DE Spearman, and Pathway activity. The Pathway activity verifier rewards cells whose pathway responses match known perturbation biology. We evaluate PerturbCellRL on multiple genetic and chemical perturbation benchmarks. Across these benchmarks, PerturbCellRL improves over the pretrained flow-matching generator on reward-aligned evaluation metrics and a held-out evaluation metric. Moreover, PerturbCellRL remains competitive with state-of-the-art methods on population-level metrics. Together, these results frame trustworthy single-cell prediction as verifier-guided generative alignment, moving beyond matching expression distributions toward predictions whose single-cell perturbation effects are explicitly checked for biological consistency.

Deep learning models are intrinsically sensitive to distribution shifts in the input data. In particular, small, barely perceivable perturbations to the input data can force models to make wrong predictions with high confidence. An common defense mechanism is regularization through adversarial training which injects worst-case perturbations back into training to strengthen the decision boundaries, and to reduce overfitting. In this context, we perform an investigation of 3x3 convolution filters that form in adversarially-trained models. Filters are extracted from 71 public models of the linf-RobustBench CIFAR-10/100 and ImageNet1k leaderboard and compared to filters extracted from models built on the same architectures but trained without robust regularization. We observe that adversarially-robust models appear to form more diverse, less sparse, and more orthogonal convolution filters than their normal counterparts. The largest differences between robust and normal models are found in the deepest layers, and the very first convolution layer, which consistently and predominantly forms filters that can partially eliminate perturbations, irrespective of the architecture. Data & Project website: https://github.com/paulgavrikov/cvpr22w_RobustnessThroughTheLens

With the increasing capabilities of large language models (LLMs), these high-performance models have achieved state-of-the-art results on a wide range of natural language processing (NLP) tasks. However, the models' performance on commonly-used benchmark datasets often fails to accurately reflect their reliability and robustness when applied to real-world noisy data. To address these challenges, we propose a unified robustness evaluation framework based on the slot-filling task to systematically evaluate the dialogue understanding capability of LLMs in diverse input perturbation scenarios. Specifically, we construct a input perturbation evaluation dataset, Noise-LLM, which contains five types of single perturbation and four types of mixed perturbation data. Furthermore, we utilize a multi-level data augmentation method (character, word, and sentence levels) to construct a candidate data pool, and carefully design two ways of automatic task demonstration construction strategies (instance-level and entity-level) with various prompt templates. Our aim is to assess how well various robustness methods of LLMs perform in real-world noisy scenarios. The experiments have demonstrated that the current open-source LLMs generally achieve limited perturbation robustness performance. Based on these experimental observations, we make some forward-looking suggestions to fuel the research in this direction.

The rapid advancement of Large Language Models (LLMs) has established standardized evaluation benchmarks as the primary instrument for model comparison. Yet, their reliability is increasingly questioned due to sensitivity to shallow variations in input prompts. This paper examines how controlled, truth-conditionally equivalent lexical and syntactic perturbations affect the absolute performance and relative ranking of 23 contemporary LLMs across three benchmarks: MMLU, SQuAD, and AMEGA. We employ two linguistically principled pipelines to generate meaning-preserving variations: one performing synonym substitution for lexical changes, and another using dependency parsing to determine applicable syntactic transformations. Results show that lexical perturbations consistently induce substantial, statistically significant performance degradation across nearly all models and tasks, while syntactic perturbations have more heterogeneous effects, occasionally improving results. Both perturbation types destabilize model leaderboards on complex tasks. Furthermore, model robustness did not consistently scale with model size, revealing strong task dependence. Overall, the findings suggest that LLMs rely more on surface-level lexical patterns than on abstract linguistic competence, underscoring the need for robustness testing as a standard component of LLM evaluation.

A variety of methods have been proposed to try to explain how deep neural networks make their decisions. Key to those approaches is the need to sample the pixel space efficiently in order to derive importance maps. However, it has been shown that the sampling methods used to date introduce biases and other artifacts, leading to inaccurate estimates of the importance of individual pixels and severely limit the reliability of current explainability methods. Unfortunately, the alternative -- to exhaustively sample the image space is computationally prohibitive. In this paper, we introduce EVA (Explaining using Verified perturbation Analysis) -- the first explainability method guarantee to have an exhaustive exploration of a perturbation space. Specifically, we leverage the beneficial properties of verified perturbation analysis -- time efficiency, tractability and guaranteed complete coverage of a manifold -- to efficiently characterize the input variables that are most likely to drive the model decision. We evaluate the approach systematically and demonstrate state-of-the-art results on multiple benchmarks.

An observable in an Effective Field Theory (EFT) is an expansion in a set of small parameters, no different than any other perturbation series. Truncating such a series expansion leaves the leading dropped term as the dominant source of error, and that term itself contains a calculable portion. We explore the partial calculation of this next-order error, available at dim-6^2 in SMEFT with no additional tools needed to simulate it, and explore how to efficiently use that partial calculation to model the next-order uncertainty in full using a minimal number of nuisance parameters. This estimate of the uncertainty of the EFT signal rate naturally imposes EFT validity by ensuring that bounds are driven by kinematic regions where truncation uncertainties are parametrically smaller than the signal. We incorporate the calculable dim-6^2 piece into the signal and cover the remaining higher-dimension uncertainty with a small set of nuisance parameters derived from a scan over Wilson-coefficient values. Our algorithm to determine the relevant nuisance parameters and distributions is automation-friendly and applies to arbitrary truncation choices. We then provide multiple examples of its implementation in high-energy collider processes focused on SMEFT truncated at O(Λ^{-2}), where the dim-6^2 piece is used to estimate the full mathcal O(Λ^{-4}) dependence. In the examples considered here the reduction of nuisance parameters is appreciable, generically reducing the nuisance parameter count by an order of magnitude compared to naïve estimates.

Large language models have revolutionized natural language processing but face significant challenges of high storage and runtime costs, due to the transformer architecture's reliance on self-attention, particularly the large Key-Value (KV) cache for long-sequence inference. Recent efforts to reduce KV cache size by pruning less critical entries based on attention weights remain empirical and lack formal grounding. This paper presents a formal study on identifying critical KV cache entries by analyzing attention output perturbation. Our analysis reveals that, beyond attention weights, the value states within KV entries and pretrained parameter matrices are also crucial. Based on this, we propose a perturbation-constrained selection algorithm that optimizes the worst-case output perturbation to identify critical entries. Evaluations on the Needle-in-a-Haystack test and Longbench benchmark show our algorithm enhances state-of-the-art cache eviction methods. Further empirical analysis confirms that our algorithm achieves lower output perturbations in over 92% attention heads in Llama model, thereby providing a significant improvement over existing methods.

Recent work found that LLMs are sensitive to a wide range of arbitrary prompt dimensions, including the type of delimiters, answer enumerators, instruction wording, and more. This throws into question popular single-prompt evaluation practices. We present DOVE (Dataset Of Variation Evaluation) a large-scale dataset containing prompt perturbations of various evaluation benchmarks. In contrast to previous work, we examine LLM sensitivity from an holistic perspective, and assess the joint effects of perturbations along various dimensions, resulting in thousands of perturbations per instance. We evaluate several model families against DOVE, leading to several findings, including efficient methods for choosing well-performing prompts, observing that few-shot examples reduce sensitivity, and identifying instances which are inherently hard across all perturbations. DOVE consists of more than 250M prompt perturbations and model outputs, which we make publicly available to spur a community-wide effort toward meaningful, robust, and efficient evaluation. Browse the data, contribute, and more: https://slab-nlp.github.io/DOVE/

Large language models (LLMs) are rapidly approaching the level of proficiency in university-level symbolic mathematics required for applications in advanced science and technology. However, existing benchmarks fall short in assessing the core skills of LLMs in symbolic mathematics-such as integration, differential equations, and algebraic simplification. To address this gap, we introduce ASyMOB, a novel assessment framework focused exclusively on symbolic manipulation, featuring 17,092 unique math challenges, organized by similarity and complexity. ASyMOB enables analysis of LLM generalization capabilities by comparing performance in problems that differ by simple numerical or symbolic `perturbations'. Evaluated LLMs exhibit substantial degradation in performance for all perturbation types (up to -70.3%), suggesting reliance on memorized patterns rather than deeper understanding of symbolic math, even among models achieving high baseline accuracy. Comparing LLM performance to computer algebra systems, we identify examples where they fail while LLMs succeed, as well as problems solved only by combining both approaches. Models capable of integrated code execution yielded higher accuracy compared to their performance without code, particularly stabilizing weaker models (up to +33.1% for certain perturbation types). Notably, the most advanced models (o4-mini, Gemini 2.5 Flash) demonstrate not only high symbolic math proficiency (scoring 96.8% and 97.6% on the unperturbed set), but also remarkable robustness against perturbations, (-21.7% and -21.2% vs. average -50.4% for the other models). This may indicate a recent "phase transition" in the generalization capabilities of frontier LLMs. It remains to be seen whether the path forward lies in deeper integration with sophisticated external tools, or in developing models so capable that symbolic math systems like CAS become unnecessary.

Combining discrete probability distributions and combinatorial optimization problems with neural network components has numerous applications but poses several challenges. We propose Implicit Maximum Likelihood Estimation (I-MLE), a framework for end-to-end learning of models combining discrete exponential family distributions and differentiable neural components. I-MLE is widely applicable as it only requires the ability to compute the most probable states and does not rely on smooth relaxations. The framework encompasses several approaches such as perturbation-based implicit differentiation and recent methods to differentiate through black-box combinatorial solvers. We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP. Moreover, we show that I-MLE simplifies to maximum likelihood estimation when used in some recently studied learning settings that involve combinatorial solvers. Experiments on several datasets suggest that I-MLE is competitive with and often outperforms existing approaches which rely on problem-specific relaxations.

GUI grounding models report over 85% accuracy on standard benchmarks, yet drop 27-56 percentage points when instructions require spatial reasoning rather than direct element naming. Current benchmarks miss this because they evaluate each screenshot once with a single fixed instruction. We introduce GUI-Perturbed, a controlled perturbation framework that independently varies visual scenes and instructions to measure grounding robustness. Evaluating three 7B models from the same architecture lineage, we find that relational instructions cause systematic accuracy collapse across all models, a 70% browser zoom produces statistically significant degradation, and rank-8 LoRA fine-tuning with augmented data degrades performance rather than improving it. By perturbing along independent axes, GUI-Perturbed isolates which specific capability axes are affected-spatial reasoning, visual robustness, reasoning calibration-providing diagnostic signal that aggregate benchmarks cannot. We release the dataset, augmentation pipeline, and a fine-tuned model.

We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs.

Recent works have suggested that finite Bayesian neural networks may sometimes outperform their infinite cousins because finite networks can flexibly adapt their internal representations. However, our theoretical understanding of how the learned hidden layer representations of finite networks differ from the fixed representations of infinite networks remains incomplete. Perturbative finite-width corrections to the network prior and posterior have been studied, but the asymptotics of learned features have not been fully characterized. Here, we argue that the leading finite-width corrections to the average feature kernels for any Bayesian network with linear readout and Gaussian likelihood have a largely universal form. We illustrate this explicitly for three tractable network architectures: deep linear fully-connected and convolutional networks, and networks with a single nonlinear hidden layer. Our results begin to elucidate how task-relevant learning signals shape the hidden layer representations of wide Bayesian neural networks.

Straight line equation y=mx with slope m, when singularly perturbed as ay^3+y=mx with a positive parameter a, results in S-shaped curves or S-curves on a real plane. As arightarrow 0, we get back y=mx which is a cumulative distribution function of a continuous uniform distribution that describes the occurrence of every event in an interval to be equally probable. As arightarrowinfty, the derivative of y has finite support only at y=0 resembling a degenerate distribution. Based on these arguments, in this work, we propose that these S-curves can represent maximum entropy uniform distribution to a zero entropy single value. We also argue that these S-curves are superposable as they are only parametrically nonlinear but fundamentally linear. So far, the superposed forms have been used to capture the patterns of natural systems such as nonlinear dynamics of biological growth and kinetics of enzyme reactions. Here, we attempt to use the S-curve and its superposed form as statistical models. We fit the models on a classical dataset containing flower measurements of iris plants and analyze their usefulness in pattern recognition. Based on these models, we claim that any non-uniform pattern can be represented as a singular perturbation to uniform distribution. However, our parametric estimation procedure have some limitations such as sensitivity to initial conditions depending on the data at hand.

Planning effective interventions in biological systems requires treatment-effect models that adapt to unseen biological contexts by identifying their specific underlying mechanisms. Yet single-cell perturbation datasets span only a handful of biological contexts, and existing methods cannot leverage new interventional evidence at inference time to adapt beyond their training data. To meta-learn a perturbation effect estimator, we present MapPFN, a prior-data fitted network (PFN) pretrained on synthetic data generated from a prior over causal perturbations. Given a set of experiments, MapPFN uses in-context learning to predict post-perturbation distributions, without gradient-based optimization. Despite being pretrained on in silico gene knockouts alone, MapPFN identifies differentially expressed genes, matching the performance of models trained on real single-cell data. Our code and data are available at https://github.com/marvinsxtr/MapPFN.

Recent image generation schemes typically capture image distribution in a pre-constructed latent space relying on a frozen image tokenizer. Though the performance of tokenizer plays an essential role to the successful generation, its current evaluation metrics (e.g. rFID) fail to precisely assess the tokenizer and correlate its performance to the generation quality (e.g. gFID). In this paper, we comprehensively analyze the reason for the discrepancy of reconstruction and generation qualities in a discrete latent space, and, from which, we propose a novel plug-and-play tokenizer training scheme to facilitate latent space construction. Specifically, a latent perturbation approach is proposed to simulate sampling noises, i.e., the unexpected tokens sampled, from the generative process. With the latent perturbation, we further propose (1) a novel tokenizer evaluation metric, i.e., pFID, which successfully correlates the tokenizer performance to generation quality and (2) a plug-and-play tokenizer training scheme, which significantly enhances the robustness of tokenizer thus boosting the generation quality and convergence speed. Extensive benchmarking are conducted with 11 advanced discrete image tokenizers with 2 autoregressive generation models to validate our approach. The tokenizer trained with our proposed latent perturbation achieve a notable 1.60 gFID with classifier-free guidance (CFG) and 3.45 gFID without CFG with a sim400M generator. Code: https://github.com/lxa9867/ImageFolder.

Privacy-preserving machine learning algorithms are crucial for the increasingly common setting in which personal data, such as medical or financial records, are analyzed. We provide general techniques to produce privacy-preserving approximations of classifiers learned via (regularized) empirical risk minimization (ERM). These algorithms are private under the ε-differential privacy definition due to Dwork et al. (2006). First we apply the output perturbation ideas of Dwork et al. (2006), to ERM classification. Then we propose a new method, objective perturbation, for privacy-preserving machine learning algorithm design. This method entails perturbing the objective function before optimizing over classifiers. If the loss and regularizer satisfy certain convexity and differentiability criteria, we prove theoretical results showing that our algorithms preserve privacy, and provide generalization bounds for linear and nonlinear kernels. We further present a privacy-preserving technique for tuning the parameters in general machine learning algorithms, thereby providing end-to-end privacy guarantees for the training process. We apply these results to produce privacy-preserving analogues of regularized logistic regression and support vector machines. We obtain encouraging results from evaluating their performance on real demographic and benchmark data sets. Our results show that both theoretically and empirically, objective perturbation is superior to the previous state-of-the-art, output perturbation, in managing the inherent tradeoff between privacy and learning performance.

Modern LLM RL systems separate rollout generation from policy optimization. These two stages are expected to produce token probabilities that match exactly. However, implementation differences can make them assign different values to the same sequence under the same model weights, inducing Training-Inference Mismatch (TIM). TIM is difficult to inspect because it is entangled with off-policy drift and common stabilization mechanisms. In this work, we isolate TIM in a zero-mismatch diagnostic setting (VeXact), and show that small token-level numerical disagreements can independently cause training collapse. We further show that TIM changes the effective optimization problem, and identify a set of remedies that could mitigate TIM. Our results suggest that TIM is not benign numerical noise, but a systems-level perturbation that should be treated as a first-order factor in analyzing LLM RL stability.

DDEBIFTOOL is a collection of Matlab routines for numerical bifurcation analysis of systems of delay differential equations with discrete constant and state-dependent delays. The package supports continuation and stability analysis of steady state solutions and periodic solutions. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. To analyse the stability of steady state solutions, approximations are computed to the rightmost, stability-determining roots of the characteristic equation which can subsequently be used as starting values in a Newton procedure. For periodic solutions, approximations to the Floquet multipliers are computed. The manual describes the structure of the package, its routines, and its data and method parameter structures.

Controlled text perturbation is useful for evaluating and improving model generalizability. However, current techniques rely on training a model for every target perturbation, which is expensive and hard to generalize. We present Tailor, a semantically-controlled text generation system. Tailor builds on a pretrained seq2seq model and produces textual outputs conditioned on control codes derived from semantic representations. We craft a set of operations to modify the control codes, which in turn steer generation towards targeted attributes. These operations can be further composed into higher-level ones, allowing for flexible perturbation strategies. We demonstrate the effectiveness of these perturbations in multiple applications. First, we use Tailor to automatically create high-quality contrast sets for four distinct natural language processing (NLP) tasks. These contrast sets contain fewer spurious artifacts and are complementary to manually annotated ones in their lexical diversity. Second, we show that Tailor perturbations can improve model generalization through data augmentation. Perturbing just 2% of training data leads to a 5.8-point gain on an NLI challenge set measuring reliance on syntactic heuristics.

Power transforms are popular parametric techniques for making data more Gaussian-like, and are widely used as preprocessing steps in statistical analysis and machine learning. However, we find that direct implementations of power transforms suffer from severe numerical instabilities, which can lead to incorrect results or even crashes. In this paper, we provide a comprehensive analysis of the sources of these instabilities and propose effective remedies. We further extend power transforms to the federated learning setting, addressing both numerical and distributional challenges that arise in this context. Experiments on real-world datasets demonstrate that our methods are both effective and robust, substantially improving stability compared to existing approaches.

Sharpness Aware Minimization (SAM) enhances performance across various neural architectures and datasets. As models are continually scaled up to improve performance, a rigorous understanding of SAM's scaling behaviour is paramount. To this end, we study the infinite-width limit of neural networks trained with SAM, using the Tensor Programs framework. Our findings reveal that the dynamics of standard SAM effectively reduce to applying SAM solely in the last layer in wide neural networks, even with optimal hyperparameters. In contrast, we identify a stable parameterization with layerwise perturbation scaling, which we call Maximal Update and Perturbation Parameterization (μP^2), that ensures all layers are both feature learning and effectively perturbed in the limit. Through experiments with MLPs, ResNets and Vision Transformers, we empirically demonstrate that μP^2 achieves hyperparameter transfer of the joint optimum of learning rate and perturbation radius across model scales. Moreover, we provide an intuitive condition to derive μP^2 for other perturbation rules like Adaptive SAM and SAM-ON, also ensuring balanced perturbation effects across all layers.

Overfitting widely exists in adversarial robust training of deep networks. An effective remedy is adversarial weight perturbation, which injects the worst-case weight perturbation during network training by maximizing the classification loss on adversarial examples. Adversarial weight perturbation helps reduce the robust generalization gap; however, it also undermines the robustness improvement. A criterion that regulates the weight perturbation is therefore crucial for adversarial training. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation. With LSC, we find that it is essential to conduct weight perturbation on adversarial data with small classification loss to eliminate robust overfitting. Weight perturbation on adversarial data with large classification loss is not necessary and may even lead to poor robustness. Based on these observations, we propose a robust perturbation strategy to constrain the extent of weight perturbation. The perturbation strategy prevents deep networks from overfitting while avoiding the side effect of excessive weight perturbation, significantly improving the robustness of adversarial training. Extensive experiments demonstrate the superiority of the proposed method over the state-of-the-art adversarial training methods.

The evaluation of Retrieval-Augmented Generation (RAG) systems typically examines retrieval quality and generation parameters like temperature in isolation, overlooking their interaction. This work presents a systematic investigation of how text perturbations (simulating noisy retrieval) interact with temperature settings across multiple LLM runs. We propose a comprehensive RAG Perturbation-Temperature Analysis Framework that subjects retrieved documents to three distinct perturbation types across varying temperature settings. Through extensive experiments on HotpotQA with both open-source and proprietary LLMs, we demonstrate that performance degradation follows distinct patterns: high-temperature settings consistently amplify vulnerability to perturbations, while certain perturbation types exhibit non-linear sensitivity across the temperature range. Our work yields three key contributions: (1) a diagnostic benchmark for assessing RAG robustness, (2) an analytical framework for quantifying perturbation-temperature interactions, and (3) practical guidelines for model selection and parameter tuning under noisy retrieval conditions.

Motivated by the need to efficiently identify multiple candidates in high trial-and-error cost tasks such as drug discovery, we propose a near-optimal algorithm to identify all ε-best arms (i.e., those at most ε worse than the optimum). Specifically, we introduce LinFACT, an algorithm designed to optimize the identification of all ε-best arms in linear bandits. We establish a novel information-theoretic lower bound on the sample complexity of this problem and demonstrate that LinFACT achieves instance optimality by matching this lower bound up to a logarithmic factor. A key ingredient of our proof is to integrate the lower bound directly into the scaling process for upper bound derivation, determining the termination round and thus the sample complexity. We also extend our analysis to settings with model misspecification and generalized linear models. Numerical experiments, including synthetic and real drug discovery data, demonstrate that LinFACT identifies more promising candidates with reduced sample complexity, offering significant computational efficiency and accelerating early-stage exploratory experiments.

In this note, we show that the Local Molecular Field theory of Weeks et. al. can be re-derived as an extremum problem for an approximate Helmholtz free energy. Using the resulting free energy as a classical, fluid density functional yields an implicit solvent method identical in form to the Molecular Density Functional theory of Borgis et. al., but with an explicit formula for the 'ideal' free energy term. This new expression for the ideal free energy term can be computed from all-atom molecular dynamics of a solvent with only short-range interactions. The key hypothesis required to make the theory valid is that all smooth (and hence long-range) energy functions obey Gaussian statistics. This is essentially a random phase approximation for perturbations from a short-range only, 'reference,' fluid. This single hypothesis is enough to prove that the self-consistent LMF procedure minimizes a novel density functional whose 'ideal' free energy is the molecular system under a specific, reference Hamiltonian, as opposed to the non-interacting gas of conventional density functionals. Implementation of this new functional into existing software should be straightforward and robust.

Predicting the effects of perturbations in-silico on cell state can identify drivers of cell behavior at scale and accelerate drug discovery. However, modeling challenges remain due to the inherent heterogeneity of single cell gene expression and the complex, latent gene dependencies. Here, we present PRiMeFlow, an end-to-end flow matching based approach to directly model the effects of genetic and small molecule perturbations in the gene expression space. The distribution-fitting approach taken by PRiMeFlow enables it to accurately approximate the empirical distribution of single-cell gene expression, which we demonstrate through extensive benchmarking inside PerturBench. Through ablation studies, we also validate important model design choices such as operating in gene expression space and parameterizing the velocity field with a U-Net architecture. Finally, by scaling PRiMeFlow to a broad perturbation data atlas spanning multiple datasets and employing a carefully designed pretraining-finetuning strategy, we demonstrate its outstanding performance on the H1 human embryonic stem cells from the ARC Virtual Cell Challenge benchmark.

We develop a general mathematical framework to analyze scaling regimes and derive explicit analytic solutions for gradient flow (GF) in large learning problems. Our key innovation is a formal power series expansion of the loss evolution, with coefficients encoded by diagrams akin to Feynman diagrams. We show that this expansion has a well-defined large-size limit that can be used to reveal different learning phases and, in some cases, to obtain explicit solutions of the nonlinear GF. We focus on learning Canonical Polyadic (CP) decompositions of high-order tensors, and show that this model has several distinct extreme lazy and rich GF regimes such as free evolution, NTK and under- and over-parameterized mean-field. We show that these regimes depend on the parameter scaling, tensor order, and symmetry of the model in a specific and subtle way. Moreover, we propose a general approach to summing the formal loss expansion by reducing it to a PDE; in a wide range of scenarios, it turns out to be first-order and solvable by the method of characteristics. We observe a very good agreement of our theoretical predictions with experimental results.

We present anyakrakusuma, an open-source Python library that solves the discrete static Schrödinger bridge problem, the entropically regularized counterpart of optimal transport, through a log-domain Sinkhorn--Knopp iteration and reconstructs the entropic interpolation between two empirical point clouds. The solver is paired with a diagnostic pipeline that characterizes the optimal coupling and the intermediate distributions through information-theoretic and geometric measures. We exercise the library on four idealized planar cases spanning a circle-to-circle dilation, a spiral-to-mixture fragmentation, a rigid reorientation of two moons, and a Lissajous-to-trefoil deformation. The log-domain formulation is necessary rather than merely convenient at the parameters studied, where the cost-to-regularization ratio reaches four hundred and the Gibbs kernel underflows double precision across most of its range; the iteration nonetheless attains a marginal residual of 10^{-9} and unit marginal fidelity in every case. Residual histories decay geometrically over approximately eight decades at per-iteration contraction factors between 0.966 and 0.976, which are local rates near the fixed point that lie many orders of magnitude below the worst-case Hilbert-metric bound. The covariance analysis recovers an imposed ninety-degree reorientation to within 0.07^circ, roughly forty times smaller than its uncertainty, across a masked interval of near-isotropy on which the principal axis is unobservable. The diagnostics are reported with explicit attention to the regimes in which each is well defined, including the differential entropy, which is meaningful only on the open interpolation interval. The presented cases are constructed rather than measured; quantitative application to empirical point clouds requires further study.

We develop a linear perturbative formalism to compute the response of an inhomogeneous stellar disk embedded in a non-responsive dark matter halo to perturbations like bars, spiral arms and satellite galaxy encounters. Without self-gravity to reinforce it, the response of a Fourier mode phase mixes away due to an intrinsic spread in the vertical (Omega_z), radial (Omega_r) and azimuthal (Omega_phi) frequencies, giving rise to local phase-space spirals. Collisional diffusion due to scattering of stars by structures like giant molecular clouds causes super-exponential damping of the phase-spiral amplitude. The z-v_z phase-spiral is 1-armed (2-armed) for vertically anti-symmetric (symmetric) bending (breathing) modes. Only transient perturbations with timescales (tau_{P}) comparable to the vertical oscillation period (tau_z sim 1/Omega_z) trigger z-v_z phase-spirals. Each (n,l,m) mode of the response to impulsive (tau_{P}<tau=1/(nOmega_z+lOmega_r+mOmega_phi)) perturbations is power law (sim tau_{P}/tau) suppressed, but that to adiabatic (tau_{P}>tau) perturbations is exponentially weak (sim left[-left(tau_{mathrm{P}/tauright)^alpharight]}) except resonant (tauto infty) modes. Slower (tau_{P}>tau_z) perturbations, e.g., distant encounters with satellite galaxies, induce stronger bending modes. If the Gaia phase-spiral was triggered by a satellite, Sagittarius is the leading contender as it dominates the Solar neighborhood response of the Milky Way disk to satellite encounters. However, survival against collisional damping necessitates that the impact occurred within sim 0.6-0.7 Gyr ago. We discuss how the detailed galactic potential dictates the phase-spiral shape: phase mixing occurs slower and phase-spirals are less wound in the outer disk and in presence of an ambient halo.

The KMOC formalism provides a systematic framework for extracting classical observables perturbatively from on-shell scattering amplitudes. In this work, we apply this formalism to compute electromagnetic observables in four dimensions, focusing in particular on the linear memory effect and its tail contributions. Using the leading and subleading soft-photon theorems to construct the soft radiation kernel, we demonstrate how these infrared observables emerge directly from amplitude data. We further show that demanding the expected non-perturbative properties of memory and tail effects imposes a nontrivial set of consistency conditions on the underlying S-matrix. We interpret these constraints as imposing the requirement of macroscopic causality on the S-matrix via analysis of inclusive observables.

Motivated by topologically protected states in wave physics, we study localised eigenmodes in one-dimensional periodic media with defects. The Su-Schrieffer-Heeger model (the canonical example of a one-dimensional system with topologically protected localised defect states) is used to demonstrate the method. Our approach can be used to describe two broad classes of perturbations to periodic differential problems: those caused by inserting a finite-sized piece of arbitrary material and those caused by creating an interface between two different periodic media. The results presented here characterise the existence of localised eigenmodes in each case and, when they exist, determine their eigenfrequencies and provide concise analytic results that quantify the decay rate of these modes. These results are obtained using both high-frequency homogenisation and transfer matrix analysis, with good agreement between the two methods.

Predicting single-cell transcriptional responses to genetic, chemical and cytokine perturbations is a fundamental challenge in computational biology and AI Virtual Cell (AIVC) modeling, with direct implications for drug discovery and the elucidation of gene regulatory networks. Existing approaches often rely on auxiliary cell-state encoders, hierarchical variational autoencoders, dedicated Transformer encoder-decoder modules, or gene-interaction priors to compress high-dimensional expression profiles into latent representations. While effective, these designs increase architectural complexity and may limit scalability and generalizability. This paper introduces OCOO-T, a minimalist flow-matching-based AIVC model for transcriptional perturbation response prediction. OCOO-T utilizes a vanilla Transformer stack that operates directly on continuous gene expression profiles and formulates perturbation response prediction as a continuous-time denoising process. Perturbation embeddings, dosage information, and cell-line/cell-type specificity are integrated through adaptive layer normalization and in-context tokens. Comprehensive evaluations on Tahoe100M, Replogle, and PBMC benchmarks demonstrate that OCOO-T achieves state-of-the-art performance across diverse perturbations and cell types while effectively scaling to long transcriptional profiles through patching and depatching of cellular contexts. By leveraging the simplicity of Transformer-based denoising for single-cell omics, OCOO-T provides an effective and scalable framework for in-silico cellular simulation.

We introduce rd-spiral, an open-source Python library for simulating 2D reaction-diffusion systems using pseudo-spectral methods. The framework combines FFT-based spatial discretization with adaptive Dormand-Prince time integration, achieving exponential convergence while maintaining pedagogical clarity. We analyze three dynamical regimes: stable spirals, spatiotemporal chaos, and pattern decay, revealing extreme non-Gaussian statistics (kurtosis >96) in stable states. Information-theoretic metrics show 10.7% reduction in activator-inhibitor coupling during turbulence versus 6.5% in stable regimes. The solver handles stiffness ratios >6:1 with features including automated equilibrium classification and checkpointing. Effect sizes (delta=0.37--0.78) distinguish regimes, with asymmetric field sensitivities to perturbations. By balancing computational rigor with educational transparency, rd-spiral bridges theoretical and practical nonlinear dynamics.

Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes mathcal{O}(d^{,3/2}/epsilon) queries to the time-evolution operator of the system and mathcal{O}(d^{,3/2}) queries to a block-encoding of the perturbation, for d cooling steps and an epsilon-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.

Existing LLM agent frameworks lack formal semantics: there is no principled way to determine whether an agent configuration is well-formed or will terminate. We present λ_A, a typed lambda calculus for agent composition that extends the simply-typed lambda calculus with oracle calls, bounded fixpoints (the ReAct loop), probabilistic choice, and mutable environments. We prove type safety, termination of bounded fixpoints, and soundness of derived lint rules, with partial Coq mechanization (1,567 lines, 43 completed proofs). As a practical application, we derive a lint tool that detects structural configuration errors directly from the operational semantics. An evaluation on 835 real-world GitHub agent configurations shows that 94.1% are structurally incomplete under λ_A, with YAML-only lint precision at 54%, rising to 96--100% under joint YAML+Python AST analysis on 175 samples. This gap quantifies, for the first time, the degree of semantic entanglement between declarative configuration and imperative code in the agent ecosystem. We further show that five mainstream paradigms (LangGraph, CrewAI, AutoGen, OpenAI SDK, Dify) embed as typed λ_A fragments, establishing λ_A as a unifying calculus for LLM agent composition.

Invertible neural networks based on coupling flows (CF-INNs) have various machine learning applications such as image synthesis and representation learning. However, their desirable characteristics such as analytic invertibility come at the cost of restricting the functional forms. This poses a question on their representation power: are CF-INNs universal approximators for invertible functions? Without a universality, there could be a well-behaved invertible transformation that the CF-INN can never approximate, hence it would render the model class unreliable. We answer this question by showing a convenient criterion: a CF-INN is universal if its layers contain affine coupling and invertible linear functions as special cases. As its corollary, we can affirmatively resolve a previously unsolved problem: whether normalizing flow models based on affine coupling can be universal distributional approximators. In the course of proving the universality, we prove a general theorem to show the equivalence of the universality for certain diffeomorphism classes, a theoretical insight that is of interest by itself.

We discuss the numerical solution of initial value problems for varepsilon^2,varphi''+a(x),varphi=0 in the highly oscillatory regime, i.e., with a(x)>0 and 0<varepsilonll 1. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude O(varepsilon^{N}) where N refers to the truncation order of the underlying asymptotic series. When the optimal truncation order N_{opt} is chosen, the error behaves like O(varepsilon^{-2}exp(-cvarepsilon^{-1})) with some c>0.

Perturbation probing generates task-specific causal hypotheses for FFN neurons in large language models using two forward passes per prompt and no backpropagation, followed by a one-time intervention sweep of about 150 passes amortized across all identified neurons. Across eight behavioral circuits, 13 models, and four architecture families, we identify two circuit structures that organize LLM behavior. Opposition circuits appear when RLHF suppresses a pre-training tendency. In safety refusal, about 50 neurons, or 0.014 percent of all neurons, control the refusal template; ablating them changes 80 percent of response formats on 520 AdvBench prompts while producing near-zero harmful compliance, 3 of 520 cases, all with disclaimers. Routing circuits appear for pre-training behaviors distributed through attention. For language selection, residual-stream direction injection switches English to Chinese output on 99.1 percent of 580 benchmark prompts in the 3 of 19 tested models that satisfy three observed conditions: bilingual training, FFN-to-skip signal ratio between 0.3 and 1.1, and linear representability. The same intervention fails on the other 16 models and on math, code, and factual circuits, defining the limits of directional steering. The FFN-to-skip signal ratio, computed from the same two forward passes, distinguishes the two structures and predicts the appropriate intervention. Circuit topology varies by architecture, from Qwen's concentrated FFN bottleneck to Gemma's normalization-shielded circuit. In Qwen3.5-2B, ablating 20 neurons eliminates multi-turn sycophantic capitulation, while amplifying 10 related neurons improves factual correction from 52 percent to 88 percent on 200 TruthfulQA prompts. These results show that perturbation probing offers mechanistic insight into RLHF-organized behavior and a practical toolkit for precision template-layer editing.

Partial Differential Equations are precise in modelling the physical, biological and graphical phenomena. However, the numerical methods suffer from the curse of dimensionality, high computation costs and domain-specific discretization. We aim to explore pros and cons of different PDE solvers, and apply them to specific scientific simulation problems, including forwarding solution, inverse problems and equations discovery. In particular, we extend the recent CNF (NeurIPS 2023) framework solver to multi-dependent-variable and non-linear settings, together with down-stream applications. The outcomes include implementation of selected methods, self-tuning techniques, evaluation on benchmark problems and a comprehensive survey of neural PDE solvers and scientific simulation applications.

Predicting how a cell will change its transcriptional state under a developmental signal or a genetic perturbation is the computational core of in-silico biology and the AI Virtual Cell program. Existing approaches either fit static control-to-treated maps that discard time, or solve multi-step ODE / Schrödinger-bridge problems on each dataset independently. We introduce Chreode, a one-step cell world model that predicts action-conditioned cell-state transitions through a structured residual transition operator. It shifts distributional evolution from inference time to training time, enabling single-pass generation while preserving a Waddington-inspired decomposition into downhill landscape flow, rotational in-tangent dynamics, and stochastic spread. The model is pretrained with a shared scVI encoder and a DiT-based dynamics backbone on a 2.4M-cell mouse embryonic atlas spanning 7 datasets. As a fine-tuning initialization, Chreode improves per-target Sinkhorn distance on Weinreb hematopoiesis and Veres islet differentiation over matched scratch models, PI-SDE, and PRESCIENT. As a transferable gene-state embedding for GEARS, the pretrained dynamics representation reduces shared-vocabulary DE20 mean squared error on Norman Perturb-seq from 0.2121 to 0.1858, a 12.4% relative improvement, without changing the GEARS training procedure. We interpret this transfer to perturbation prediction as evidence that pretrained developmental-trajectory dynamics encode differentiation primitives transferable to CRISPR-induced state shifts, since both involve cell-state transitions in a shared latent geometry. The pretrained backbone additionally produces zero-shot clonal fate scores on Weinreb that are competitive with strong dynamic-OT baselines.

We study how the next-token prediction of an autoregressive Transformer language model changes under small perturbations of earlier input token embeddings. Motivated by operator learning and iterative solvers for differential equations, we investigate how the influence of one token on another decays with distance in a trained model. In multilevel methods for differential equations, such as domain decomposition, multigrid, and multilevel preconditioning, one often exploits a separation between strong local interactions and weaker but essential global interactions. The latter correspond to the long tail of the Green's function and are typically handled by a coarse-level operator. Inspired by this perspective, we compute an empirical, distance-resolved gradient profile of token dependencies using autograd. Experiments on trained Pythia models and Qwen2.5-0.5B show that, over the measured distance range, the median Jacobian sensitivity is much better described by a power-law-type decay than by an exponential alternative: the diagonal-normalized profile is well described by $overline G(r) approx γ+β(r+1)^{-p} with exponents p \approx 0.7--0.9 (typically 0.8--0.9$). This behavior appears on coherent text from Gutenberg and WikiText-103. Token-shuffling experiments show that the power-law profile persists even when syntax and prediction quality collapse, whereas randomly initialized models do not exhibit it. The slowly decaying long-range sensitivity thus appears to be a learned property of trained autoregressive Transformer operators. These findings suggest that hierarchical or coarse-level mechanisms in language models may be able to exploit the long-tailed sensitivity profiles.

Learning distribution families over R^d is a fundamental problem in unsupervised learning and statistics. A central question in this setting is whether a given family of distributions possesses sufficient structure to be (at least) information-theoretically learnable and, if so, to characterize its sample complexity. In 2018, Ashtiani et al. reframed sample compressibility, originally due to Littlestone and Warmuth (1986), as a structural property of distribution classes, proving that it guarantees PAC-learnability. This discovery subsequently enabled a series of recent advancements in deriving nearly tight sample complexity bounds for various high-dimensional open problems. It has been further conjectured that the converse also holds: every learnable class admits a tight sample compression scheme. In this work, we establish that sample compressible families remain learnable even from perturbed samples, subject to a set of necessary and sufficient conditions. We analyze two models of data perturbation: (i) an additive independent noise model, and (ii) an adversarial corruption model, where an adversary manipulates a limited subset of the samples unknown to the learner. Our results are general and rely on as minimal assumptions as possible. We develop a perturbation-quantization framework that interfaces naturally with the compression scheme and leads to sample complexity bounds that scale gracefully with the noise level and corruption budget. As concrete applications, we establish new sample complexity bounds for learning finite mixtures of high-dimensional uniform distributions under both noise and adversarial perturbations, as well as for learning Gaussian mixture models from adversarially corrupted samples, resolving two open problems in the literature.

Pre-trained large language models (LLMs) require fine-tuning to improve their responsiveness to natural language instructions. Federated learning (FL) offers a way to perform fine-tuning using the abundant data on end devices without compromising data privacy. Most existing federated fine-tuning methods for LLMs rely on parameter-efficient fine-tuning techniques, which may not reach the performance heights possible with full-parameter tuning. However, the communication overhead associated with full-parameter tuning is prohibitively high for both servers and clients. This work introduces FedKSeed, a novel approach that employs zeroth-order optimization (ZOO) with a set of random seeds. It enables federated full-parameter tuning of billion-sized LLMs directly on devices. Our method significantly reduces transmission requirements between the server and clients to just a few scalar gradients and random seeds, amounting to only a few thousand bytes. Building on this, we develop a strategy to assess the significance of ZOO perturbations for FL, allowing for probability-differentiated seed sampling. This prioritizes perturbations that have a greater impact on model accuracy. Experiments across six scenarios with different LLMs, datasets and data partitions demonstrate that our approach outperforms existing federated LLM fine-tuning methods in terms of both communication efficiency and new task generalization.

Predicting how cells respond to genetic perturbations is fundamental to understanding gene function, disease mechanisms, and therapeutic development. While recent deep learning approaches have shown promise in modeling single-cell perturbation responses, they struggle to generalize across cell types and perturbation contexts due to limited contextual information during generation. We introduce PT-RAG (Perturbation-aware Two-stage Retrieval-Augmented Generation), a novel framework that extends Retrieval-Augmented Generation beyond traditional language-model applications to cellular biology. Unlike standard RAG systems designed for text retrieval with pre-trained LLMs, perturbation retrieval lacks established similarity metrics and requires learning what constitutes relevant context, making differentiable retrieval essential. PT-RAG addresses this through a two-stage pipeline: first, retrieving candidate perturbations K using GenePT embeddings, then adaptively refining the selection through Gumbel-Softmax discrete sampling conditioned on both the cell state and the input perturbation. This cell-type-aware differentiable retrieval enables end-to-end optimization of the retrieval objective jointly with generation. On the Replogle-Nadig single-gene perturbation dataset, we demonstrate that PT-RAG outperforms both STATE and vanilla RAG under identical experimental conditions, with the strongest gains in distributional similarity metrics (W_1, W_2). Notably, vanilla RAG's dramatic failure is itself a key finding: it demonstrates that differentiable, cell-type-aware retrieval is essential in this domain, and that naive retrieval can actively harm performance. Our results establish retrieval-augmented generation as a promising paradigm for modelling cellular responses to gene perturbation. The code to reproduce our experiments is available at https://github.com/difra100/PT-RAG_ICLR.

Large language models (LLMs) are widely used as information sources, yet small changes in semantic assumptions can destabilize their beliefs. We introduce P-StaT (Perturbation Stability of Truth), a framework for evaluating belief stability under matched semantic perturbations in both representational and behavioral settings. Across 21 LLMs and three domains, we compare perturbations involving familiar Fictional statements against synthetically generated and unfamiliar Synthetic statements. Unfamiliar Synthetic perturbations generally induce greater epistemic instability than familiar Fictional perturbations, with behavioral belief retraction rates frequently exceeding 0.5 (50%). Finally, exploratory clustering analyses reveal recurring themes among retracted statements, including ambiguity, technical terminology, and obscure concepts. These results show that epistemic familiarity is systematically associated with stability under semantic reframing, suggesting that stability-based analyses can complement accuracy-based benchmarks when evaluating LLM robustness and reliability. Code and data are available at https://github.com/samanthadies/P-StaT.

We present an open-source Python implementation of an idealized high-order pseudo-spectral solver for the one-dimensional nonlinear Schr\"odinger equation (NLSE). The solver combines Fourier spectral spatial discretization with an adaptive eighth-order Dormand-Prince time integration scheme to achieve machine-precision conservation of mass and near-perfect preservation of momentum and energy for smooth solutions. The implementation accurately reproduces fundamental NLSE phenomena including soliton collisions with analytically predicted phase shifts, Akhmediev breather dynamics, and the development of modulation instability from noisy initial conditions. Four canonical test cases validate the numerical scheme: single soliton propagation, two-soliton elastic collision, breather evolution, and noise-seeded modulation instability. The solver employs a 2/3 dealiasing rule with exponential filtering to prevent aliasing errors from the cubic nonlinearity. Statistical analysis using Shannon, R\'enyi, and Tsallis entropies quantifies the spatio-temporal complexity of solutions, while phase space representations reveal the underlying coherence structure. The implementation prioritizes code transparency and educational accessibility over computational performance, providing a valuable pedagogical tool for exploring nonlinear wave dynamics. Complete source code, documentation, and example configurations are freely available, enabling reproducible computational experiments across diverse physical contexts where the NLSE governs wave evolution, including nonlinear optics, Bose-Einstein condensates, and ocean surface waves.

Reasoning failures in large language models (LLMs) are typically measured only at the end of a generation, yet many failures manifest as a process-level breakdown: the model "loses the thread" mid-reasoning. We study whether such breakdowns are detectable from inference-time observables available in standard APIs (token log probabilities), without any training or fine-tuning. We define a simple instability signal that combines consecutive-step distributional shift (JSD) and uncertainty (entropy), summarize each trace by its peak instability strength, and show that this signal reliably predicts failure. Across GSM8K and HotpotQA, instability strength predicts wrong answers with above-chance AUC and yields monotonic bucket-level accuracy decline at scale across model sizes. Crucially, we show that instability is not uniformly harmful: early instability can reflect subsequent stabilization and a correct final answer (corrective instability), whereas late instability is more often followed by failure (destructive instability), even at comparable peak magnitudes, indicating that recoverability depends not only on how strongly the distribution changes but also on when such changes occur relative to the remaining decoding horizon. The method is model-agnostic, training-free, and reproducible, and is presented as a diagnostic lens rather than a corrective or control mechanism.

We revisit the standard perturbation-based approach of Abernethy et al. (2008) in the context of unconstrained Bandit Linear Optimization (uBLO). We show the surprising result that in the unconstrained setting, this approach effectively reduces Bandit Linear Optimization (BLO) to a standard Online Linear Optimization (OLO) problem. Our framework improves on prior work in several ways. First, we derive expected-regret guarantees when our perturbation scheme is combined with comparator-adaptive OLO algorithms, leading to new insights about the impact of different adversarial models on the resulting comparator-adaptive rates. We also extend our analysis to dynamic regret, obtaining the first guarantees with optimal P_T path-length dependencies without prior knowledge of P_T. We then develop the first high-probability guarantees for both static and dynamic regret in uBLO. Finally, we discuss lower bounds on the static regret, and prove the folklore Ω(dT) rate for adversarial linear bandits on the Euclidean ball, which is of independent interest.

In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

Training Large Language Models (LLMs) with synthetic data is a prevalent practice in code generation. A key approach is self-training, where LLMs are iteratively trained on self-generated correct code snippets. In this case, the self-generated codes are drawn from a conditional distribution, conditioned on a specific seed description. However, the seed description is not the only valid representation that aligns with its intended meaning. With all valid descriptions and codes forming a joint space, codes drawn from the conditional distribution would lead to an underrepresentation of the full description-code space. As such, we propose Gibbs Fine-Tuning (GiFT), a novel self-training method inspired by Gibbs sampling. GiFT allows self-generated data to be drawn from the marginal distribution of the joint space, thereby mitigating the biases inherent in conditional sampling. We provide a theoretical analysis demonstrating the potential benefits of fine-tuning LLMs with code derived from the marginal distribution. Furthermore, we propose a perplexity-based code selection method to mitigate the imbalanced long-tail distribution of the self-generated codes. Empirical evaluation of two LLMs across four datasets demonstrates that GiFT achieves superior performance, particularly on more challenging benchmarks. Source code is available at https://github.com/Alex-HaochenLi/GiFT.

Ensemble Kalman Inversion (EKI) is a derivative-free, ensemble-based method for inverse and optimization problems. Its continuous-time formulation can be interpreted as an interacting particle system driven by a Kalman-type preconditioned descent direction. A well-known limitation of this dynamics is the possible premature collapse of the covariance of the ensemble, which makes the method sensitive to the initial ensemble. We introduce a second-order particle system in which the particles evolve according to an inertial dynamics. The model combines a Kalman-type relaxation force with damping, attraction towards the ensemble mean, and a short-range repulsive interaction designed to counteract ensemble collapse. The resulting dynamics can be interpreted as a heavy-ball reformulation of continuous-time EKI enriched by competing attractive and repulsive mechanisms. For linear inverse problems, we analyze the induced mean and fluctuation dynamics and identify a parameter regime in which fully collapsed configurations are linearly unstable. We further characterize asymptotic equilibria through a constrained optimality condition on the subspace retained by the limiting ensemble covariance and derive an exponential decay estimate. Numerical experiments illustrate the effect of inertia and repulsion on the ensemble dynamics and compare the proposed second-order method with first-order EKI-type

Large Language Models (LLMs) are widely adopted for assisting in software development tasks, yet their performance evaluations have narrowly focused on the functional correctness of generated code. Human programmers, however, require LLM-generated code to be not only correct but also optimally efficient. We propose PerfCodeGen, a training-free framework that enhances the performance of LLM-generated code by incorporating feedback based on runtime during test case execution into the self-refinement iterations. With PerfCodeGen, we achieve speedups for a significantly higher proportion of problems compared to using the base LLM with sophisticated prompting techniques. Applied to open language models like Phi-3-mini, PerfCodeGen achieves runtime efficiency comparable to prompting powerful closed models like GPT-4. We achieve state-of-the-art runtime efficiency on benchmarks such as HumanEval, MBPP, and APPS, frequently surpassing the ground truth reference solutions with PerfCodeGen using GPT-3.5 and GPT-4. Additionally, we demonstrate the effectiveness of our approach in enhancing code quality across a range of open LLMs of varying sizes including Phi-3-mini, Llama 3 8B, Mixtral 8x7B, Command R, and Llama 3 70B.

Classifier-free guidance (CFG) has become an essential component of modern diffusion models to enhance both generation quality and alignment with input conditions. However, CFG requires specific training procedures and is limited to conditional generation. To address these limitations, we propose Token Perturbation Guidance (TPG), a novel method that applies perturbation matrices directly to intermediate token representations within the diffusion network. TPG employs a norm-preserving shuffling operation to provide effective and stable guidance signals that improve generation quality without architectural changes. As a result, TPG is training-free and agnostic to input conditions, making it readily applicable to both conditional and unconditional generation. We further analyze the guidance term provided by TPG and show that its effect on sampling more closely resembles CFG compared to existing training-free guidance techniques. Extensive experiments on SDXL and Stable Diffusion 2.1 show that TPG achieves nearly a 2times improvement in FID for unconditional generation over the SDXL baseline, while closely matching CFG in prompt alignment. These results establish TPG as a general, condition-agnostic guidance method that brings CFG-like benefits to a broader class of diffusion models. The code is available at https://github.com/TaatiTeam/Token-Perturbation-Guidance

High-throughput screening techniques, such as microscopy imaging of cellular responses to genetic and chemical perturbations, play a crucial role in drug discovery and biomedical research. However, robust perturbation screening for de novo cell lines remains challenging due to the significant morphological and biological heterogeneity across cell lines. To address this, we propose a novel framework that integrates external biological knowledge into existing pretraining strategies to enhance microscopy image profiling models. Our approach explicitly disentangles perturbation-specific and cell line-specific representations using external biological information. Specifically, we construct a knowledge graph leveraging protein interaction data from STRING and Hetionet databases to guide models toward perturbation-specific features during pretraining. Additionally, we incorporate transcriptomic features from single-cell foundation models to capture cell line-specific representations. By learning these disentangled features, our method improves the generalization of imaging models to de novo cell lines. We evaluate our framework on the RxRx database through one-shot fine-tuning on an RxRx1 cell line and few-shot fine-tuning on cell lines from the RxRx19a dataset. Experimental results demonstrate that our method enhances microscopy image profiling for de novo cell lines, highlighting its effectiveness in real-world phenotype-based drug discovery applications.

Preference learning in large language models relies on reward models as proxies for human judgment. However, these models frequently exhibit preference instability, producing contradictory preference assignments in response to subtle, meaning-preserving input variations. We analyze this instability at the representation level under three semantic-preserving perturbation types: paraphrasing, pattern injection, and backdoor triggers. We attribute this instability to over-reliance on predictive yet brittle features, which we term unstable features, and isolate them via Sparse Autoencoders (SAEs) in a sparse latent space where benign and perturbed inputs activate distinctly separable patterns. Building on this separability, we propose two SAE-based instability mitigation strategies: SAE Feature Steering, which identifies and suppresses anomalously activated features at inference, and SAE Residual Correction, which learns adaptive adjustments over SAE features to restore correct preferences. Our methods substantially reduce incorrect preference assignments on harmlessness and hallucination benchmarks while preserving benign performance and general utility on other tasks, without retraining the reward model. Our code and data are available in https://github.com/shunchang-liu/pisa.

Perturbation-based explanation methods often measure the contribution of an input feature to an image classifier's outputs by heuristically removing it via e.g. blurring, adding noise, or graying out, which often produce unrealistic, out-of-samples. Instead, we propose to integrate a generative inpainter into three representative attribution methods to remove an input feature. Our proposed change improved all three methods in (1) generating more plausible counterfactual samples under the true data distribution; (2) being more accurate according to three metrics: object localization, deletion, and saliency metrics; and (3) being more robust to hyperparameter changes. Our findings were consistent across both ImageNet and Places365 datasets and two different pairs of classifiers and inpainters.

The solution of linear inverse problems arising, for example, in signal and image processing is a challenging problem since the ill-conditioning amplifies, in the solution, the noise present in the data. Recently introduced algorithms based on deep learning overwhelm the more traditional model-based approaches in performance, but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyze the trade-off between stability and accuracy of neural networks, when used to solve linear imaging inverse problems for not under-determined cases. Moreover, we propose different supervised and unsupervised solutions to increase the network stability and maintain a good accuracy, by means of regularization properties inherited from a model-based iterative scheme during the network training and pre-processing stabilizing operator in the neural networks. Extensive numerical experiments on image deblurring confirm the theoretical results and the effectiveness of the proposed deep learning-based approaches to handle noise on the data.

Nonlinear ordinary differential equations (ODEs) are powerful tools for modeling real-world dynamical systems. However, propagating initial state uncertainty through nonlinear dynamics, especially when the ODE is unknown and learned from data, remains a major challenge. This paper introduces a novel continuum dynamics perspective for model learning that enables formal uncertainty propagation by constructing Taylor series approximations of probabilistic events. We establish sufficient conditions for the soundness of the approach and prove its asymptotic convergence. Empirical results demonstrate the framework's effectiveness, particularly when predicting rare events.

We consider an existing conjecture addressing the asymptotic behavior of neural networks in the large width limit. The results that follow from this conjecture include tight bounds on the behavior of wide networks during stochastic gradient descent, and a derivation of their finite-width dynamics. We prove the conjecture for deep networks with polynomial activation functions, greatly extending the validity of these results. Finally, we point out a difference in the asymptotic behavior of networks with analytic (and non-linear) activation functions and those with piecewise-linear activations such as ReLU.

Explaining predictions based on multivariate time series data carries the additional difficulty of handling not only multiple features, but also time dependencies. It matters not only what happened, but also when, and the same feature could have a very different impact on a prediction depending on this time information. Previous work has used perturbation-based saliency methods to tackle this issue, perturbing an input using a trainable mask to discover which features at which times are driving the predictions. However these methods introduce fixed perturbations, inspired from similar methods on static data, while there seems to be little motivation to do so on temporal data. In this work, we aim to explain predictions by learning not only masks, but also associated perturbations. We empirically show that learning these perturbations significantly improves the quality of these explanations on time series data.

Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data-fidelity term, added at a few points in the domain to the standard residual and boundary losses. We show that this supervision term acts as an operator-level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, independently of how the pointwise labels are obtained. For a broad class of PDEs admitting a Feynman-Kac (FK) representation, we generate such labels by Monte Carlo averages of the FK functional, resulting in what we call ``FK-PINNs", and using the excess risk decomposition approach, we derive non-asymptotic L^2(Ω)-error bounds for FK-PINNs with tanh activation trained by finitely many steps of gradient descent. Along the way, we establish pseudo-dimension bounds for first- and second-order derivatives of tanh neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schrödinger, mean exit time, and committor problems corroborate the theory, and show that FK-PINNs can successfully solve PDEs for which standard PINNs exhibit severe failure modes.

This work provides a nonasymptotic error analysis of quantum Krylov algorithms based on real-time evolutions, subject to generic errors in the outputs of the quantum circuits. We prove upper and lower bounds on the resulting ground state energy estimates, and the error associated to the upper bound is linear in the input error rates. This resolves a misalignment between known numerics, which exhibit approximately linear error scaling, and prior theoretical analysis, which only provably obtained scaling with the error rate to the power 2{3}. Our main technique is to express generic errors in terms of an effective target Hamiltonian studied in an effective Krylov space. These results provide a theoretical framework for understanding the main features of quantum Krylov errors.

Instruction tuning has recently been recognized as an effective way of aligning Large Language Models (LLMs) to enhance their generalization ability across various tasks. However, when tuning publicly accessible, centralized LLMs with private instruction data, privacy concerns are inevitable. While direct transfer of parameterized modules between models is a plausible approach to address this, its implications and effectiveness need further exploration. This paper focuses on Offsite-Tuning (OFT), a representative technique that transfers transformer blocks between centralized LLMs and downstream emulators. Given the limited understanding of the underlying mechanism of OFT, we perform an empirical analysis on LLMs from the perspectives of representation and functional similarity. Interestingly, our findings reveal a unique modular structure within the layers of LLMs that appears to emerge as the model size expands. Simultaneously, we note subtle but potentially significant changes in representation and intermediate predictions across the layers. Inspired by these observations, we propose CRaSh, involving Clustering, Removing, and Sharing, a training-free strategy to derive improved emulators from LLMs. CRaSh significantly boosts performance of OFT with billions of parameters. Furthermore, we investigate the optimal solutions yielded by fine-tuning with and without full model through the lens of loss landscape. Our findings demonstrate a linear connectivity among these optima falling over the same basin, thereby highlighting the effectiveness of CRaSh and OFT. The source code is publicly available at https://github.com/TsinghuaC3I/CRaSh.

We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. When applied to graph Laplacians, this leads to ultra-sparsifiers that in expectation behave as the nearly-optimal ones given by [Kolla-Makarychev-Saberi-Teng STOC`10]. Combining this with the recursive preconditioning framework by [Spielman-Teng STOC`04] and improved embedding algorithms, this leads to algorithms that solve symmetric diagonally dominant linear systems and electrical flow problems in expected time close to mlog^{1/2}n .

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

Neural PDE solvers are increasingly used as learned surrogates for families of partial differential equations, where the key machine learning challenge is not only interpolation on a fixed benchmark distribution but generalization under structured shifts in coefficients, boundary conditions, discretization, and rollout horizon. Yet evaluation is still often dominated by in-distribution test error, making robustness difficult to assess. We introduce a standardized stress-testing framework for neural PDE solvers under deployment-relevant shift. We instantiate it on three representative architectures -- Fourier Neural Operators (FNOs), a DeepONet-style model, and convolutional neural operators (CNOs) -- across five qualitatively different PDE families: dispersive, elliptic, multi-scale fluid, financial, and chaotic systems. Across 750 trained models, we measure robustness using baseline-normalized degradation factors together with spectral and rollout diagnostics. The resulting comparisons reveal that strong in-distribution accuracy does not reliably predict robustness, and that failure patterns depend jointly on architecture and PDE family. Our results provide a clearer basis for evaluating robustness claims in neural PDE solvers and suggest that function-space generalization under structured shift should be treated as a first-class evaluation target.

Designing sampling algorithms for multimodal targets that remain stable under refinement of the finite-dimensional approximation of an underlying function-space problem is a central challenge. Annealed Langevin dynamics (ALD) is a natural alternative to classical Langevin in this context, since it is often observed to improve exploration across modes. Yet a gap remains between its empirical success and existing theory: under which conditions can ALD be guaranteed to remain stable across dimensions? In this paper, we bridge this gap by providing a uniform-in-dimension analysis of continuous-time ALD for Gaussian-mixture targets. Along an explicit annealing path obtained by gradually removing Gaussian smoothing from the target, we identify spectral conditions linking the smoothing covariance to the component covariances under which ALD achieves a prescribed accuracy in Kullback-Leibler divergence within a dimension-uniform time horizon. We then establish stability in a perturbative regime with imperfect initialization and approximate scores. Under a misspecified-mixture score model, we show that preconditioning ALD with an operator whose spectrum decays sufficiently fast prevents error terms from accumulating across coordinates and thereby preserves dimension-uniform control.

We introduce gridfm-datakit-v1, a Python library for generating realistic and diverse Power Flow (PF) and Optimal Power Flow (OPF) datasets for training Machine Learning (ML) solvers. Existing datasets and libraries face three main challenges: (1) lack of realistic stochastic load and topology perturbations, limiting scenario diversity; (2) PF datasets are restricted to OPF-feasible points, hindering generalization of ML solvers to cases that violate operating limits (e.g., branch overloads or voltage violations); and (3) OPF datasets use fixed generator cost functions, limiting generalization across varying costs. gridfm-datakit addresses these challenges by: (1) combining global load scaling from real-world profiles with localized noise and supporting arbitrary N-k topology perturbations to create diverse yet realistic datasets; (2) generating PF samples beyond operating limits; and (3) producing OPF data with varying generator costs. It also scales efficiently to large grids (up to 10,000 buses). Comparisons with OPFData, OPF-Learn, PGLearn, and PFΔ are provided. Available on GitHub at https://github.com/gridfm/gridfm-datakit under Apache 2.0 and via `pip install gridfm-datakit`.

We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Throughout, we show how the analysis extends to AD methods for computing higher order derivatives using a Taylor approximation.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

Ensembles of neural network weight matrices are studied through the training process for the MNIST classification problem, testing the efficacy of matrix models for representing their distributions, under assumptions of Gaussianity and permutation-symmetry. The general 13-parameter permutation invariant Gaussian matrix models are found to be effective models for the correlated Gaussianity in the weight matrices, beyond the range of applicability of the simple Gaussian with independent identically distributed matrix variables, and notably well beyond the initialisation step. The representation theoretic model parameters, and the graph-theoretic characterisation of the permutation invariant matrix observables give an interpretable framework for the best-fit model and for small departures from Gaussianity. Additionally, the Wasserstein distance is calculated for this class of models and used to quantify the movement of the distributions over training. Throughout the work, the effects of varied initialisation regimes, regularisation, layer depth, and layer width are tested for this formalism, identifying limits where particular departures from Gaussianity are enhanced and how more general, yet still highly-interpretable, models can be developed.

Although model editing has shown promise in revising knowledge in Large Language Models (LLMs), its impact on the inherent capabilities of LLMs is often overlooked. In this work, we reveal a critical phenomenon: even a single edit can trigger model collapse, manifesting as significant performance degradation in various benchmark tasks. However, benchmarking LLMs after each edit, while necessary to prevent such collapses, is impractically time-consuming and resource-intensive. To mitigate this, we propose using perplexity as a surrogate metric, validated by extensive experiments demonstrating changes in an edited model's perplexity are strongly correlated with its downstream task performances. We further conduct an in-depth study on sequential editing, a practical setting for real-world scenarios, across various editing methods and LLMs, focusing on hard cases from our previous single edit studies. The results indicate that nearly all examined editing methods result in model collapse after only few edits. To facilitate further research, we have utilized GPT-3.5 to develop a new dataset, HardEdit, based on those hard cases. This dataset aims to establish the foundation for pioneering research in reliable model editing and the mechanisms underlying editing-induced model collapse. We hope this work can draw the community's attention to the potential risks inherent in model editing practices.

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to train approximations to multiple differential equations simultaneously and are thus a general purpose solver that can be adapted to downstream tasks. Current PDE foundation models focus on either learning general solution operators and/or the governing system of equations, and thus only handle numerical or symbolic modalities. However, real-world applications may require more flexible data modalities, e.g. text analysis or descriptive outputs. To address this gap, we propose a novel multimodal deep learning approach that leverages a transformer-based architecture to approximate solution operators for a wide variety of ODEs and PDEs. Our method integrates numerical inputs, such as equation parameters and initial conditions, with text descriptions of physical processes or system dynamics. This enables our model to handle settings where symbolic representations may be incomplete or unavailable. In addition to providing accurate numerical predictions, our approach generates interpretable scientific text descriptions, offering deeper insights into the underlying dynamics and solution properties. The numerical experiments show that our model provides accurate solutions for in-distribution data (with average relative error less than 3.3%) and out-of-distribution data (average relative error less than 7.8%) together with precise text descriptions (with correct descriptions generated 100% of times). In certain tests, the model is also shown to be capable of extrapolating solutions in time.

When training large-scale models, the performance typically scales with the number of parameters and the dataset size according to a slow power law. A fundamental theoretical and practical question is whether comparable performance can be achieved with significantly smaller models and substantially less data. In this work, we provide a positive and constructive answer. We prove that a generic permutation-invariant function of d objects can be asymptotically compressed into a function of polylog d objects with vanishing error, which is proved to be the optimal compression rate. This theorem yields two key implications: (Ia) a large neural network can be compressed to polylogarithmic width while preserving its learning dynamics; (Ib) a large dataset can be compressed to polylogarithmic size while leaving the loss landscape of the corresponding model unchanged. Implication (Ia) directly establishes a proof of the dynamical lottery ticket hypothesis, which states that any ordinary network can be strongly compressed such that the learning dynamics and result remain unchanged. (Ib) shows that a neural scaling law of the form Lsim d^{-α} can be boosted to an arbitrarily fast power law decay, and ultimately to exp(-α' sqrt[m]{d}).

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.

Open quantum systems lose energy and phase coherence through different dissipative processes, but these processes can produce overlapping dynamical signatures. The Liouvillian spectrum summarizes how such a system relaxes, yet it is not obvious how much of that spectrum is needed to distinguish the underlying dissipation rates. We study this question for amplitude damping and dephasing in a six-qubit Lindblad model whose spectrum can be derived analytically. We retain only the slowest non-steady spectral modes and ask how many are required before each dissipative rate becomes recoverable. We show that population modes contain no dephasing information, which creates a lower bound of D = 2^n retained modes for uniform dephasing identifiability in the relevant rate regime. The measured recovery threshold reaches this bound at n = 4,5,6, while n = 3 remains above it. At n = 6, least squares achieves a mean joint absolute error of order 10^-9, compared with 4.355 x 10^-4 for four tabular learning methods. Robustness tests show that this advantage weakens when the spectra are perturbed and when a transverse field breaks the commuting structure. These results show that the amount and structure of retained spectral information can determine whether dissipative parameters are recoverable, independently of the estimator used. The present conclusions apply to noise-free simulator spectra rather than measurement-derived spectra.

Grokking, the sudden generalization that occurs after prolonged overfitting, is a surprising phenomenon challenging our understanding of deep learning. Although significant progress has been made in understanding grokking, the reasons behind the delayed generalization and its dependence on regularization remain unclear. In this work, we argue that without regularization, grokking tasks push models to the edge of numerical stability, introducing floating point errors in the Softmax function, which we refer to as Softmax Collapse (SC). We demonstrate that SC prevents grokking and that mitigating SC enables grokking without regularization. Investigating the root cause of SC, we find that beyond the point of overfitting, the gradients strongly align with what we call the na\"ive loss minimization (NLM) direction. This component of the gradient does not alter the model's predictions but decreases the loss by scaling the logits, typically by scaling the weights along their current direction. We show that this scaling of the logits explains the delay in generalization characteristic of grokking and eventually leads to SC, halting further learning. To validate our hypotheses, we introduce two key contributions that address the challenges in grokking tasks: StableMax, a new activation function that prevents SC and enables grokking without regularization, and perpGrad, a training algorithm that promotes quick generalization in grokking tasks by preventing NLM altogether. These contributions provide new insights into grokking, elucidating its delayed generalization, reliance on regularization, and the effectiveness of existing grokking-inducing methods. Code for this paper is available at https://github.com/LucasPrietoAl/grokking-at-the-edge-of-numerical-stability.

We introduce Power Bundle Adjustment as an expansion type algorithm for solving large-scale bundle adjustment problems. It is based on the power series expansion of the inverse Schur complement and constitutes a new family of solvers that we call inverse expansion methods. We theoretically justify the use of power series and we prove the convergence of our approach. Using the real-world BAL dataset we show that the proposed solver challenges the state-of-the-art iterative methods and significantly accelerates the solution of the normal equation, even for reaching a very high accuracy. This easy-to-implement solver can also complement a recently presented distributed bundle adjustment framework. We demonstrate that employing the proposed Power Bundle Adjustment as a sub-problem solver significantly improves speed and accuracy of the distributed optimization.

Visual Language Models (VLMs) have achieved remarkable progress, yet their reliability under small, meaning-preserving input changes remains poorly understood. We present the first large-scale, systematic study of VLM robustness to benign visual and textual perturbations: pixel-level shifts, light geometric transformations, padded rescaling, paraphrasing, and multilingual rewrites that do not alter the underlying semantics of an image-question pair. Across a broad set of models and datasets, we find that modern VLMs are highly sensitive to such minor perturbations: a substantial fraction of samples change their predicted answer under at least one visual or textual modification. We characterize how this instability varies across perturbation types, question categories, and models, revealing that even state-of-the-art systems (e.g., GPT-4o, Gemini 2.0 Flash) frequently fail under shifts as small as a few pixels or harmless rephrasings. We further show that sample-level stability serves as a strong indicator of correctness: stable samples are consistently far more likely to be answered correctly. Leveraging this, we demonstrate that the stability patterns of small, accessible open-source models can be used to predict the correctness of much larger closed-source models with high precision. Our findings expose a fundamental fragility in current VLMs and highlight the need for robustness evaluations that go beyond adversarial perturbations, focusing instead on invariances that models should reliably uphold.

We develop an asymptotic perturbation theory for the large logarithmic behavior of the non-linear integro-differential equation describing the soft correlations of QCD jet measurements, the Banfi-Marchesini-Smye (BMS) equation. This equation captures the late-time evolution of radiating color dipoles after a hard collision. This allows us to prove that at large values of the control variable (the non-global logarithm, a function of the infra-red energy scales associated with distinct hard jets in an event), the distribution has a gaussian tail. We compute the decay width analytically, giving a closed form expression, and find it to be jet geometry independent, up to the number of legs of the dipole in the active jet. Enabling the asymptotic expansion is the correct perturbative seed, where we perturb around an anzats encoding formally no real emissions, an intuition motivated by the buffer region found in jet dynamics. This must be supplemented with the correct application of the BFKL approximation to the BMS equation in collinear limits. Comparing to the asymptotics of the conformally related evolution equation encountered in small-x physics, the Balitisky-Kovchegov (BK) equation, we find that the asymptotic form of the non-global logarithms directly maps to the black-disc unitarity limit of the BK equation, despite the contrasting physical pictures. Indeed, we recover the equations of saturation physics in the final state dynamics of QCD.

Recently, Multimodal Large Language Models (MLLMs) have gained significant attention across various domains. However, their widespread adoption has also raised serious safety concerns. In this paper, we uncover a new safety risk of MLLMs: the output preference of MLLMs can be arbitrarily manipulated by carefully optimized images. Such attacks often generate contextually relevant yet biased responses that are neither overtly harmful nor unethical, making them difficult to detect. Specifically, we introduce a novel method, Preference Hijacking (Phi), for manipulating the MLLM response preferences using a preference hijacked image. Our method works at inference time and requires no model modifications. Additionally, we introduce a universal hijacking perturbation -- a transferable component that can be embedded into different images to hijack MLLM responses toward any attacker-specified preferences. Experimental results across various tasks demonstrate the effectiveness of our approach. The code for Phi is accessible at https://github.com/Yifan-Lan/Phi.

We present semantic correctness proofs of Automatic Differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Finally, we sketch how the analysis extends to other AD methods by considering a continuation-based method.

Spiking Neural Networks (SNNs) offer energy-efficient, biologically plausible computation but suffer from non-differentiable spike generation, necessitating reliance on heuristic surrogate gradients. This paper introduces UltraLIF, a principled framework that replaces surrogate gradients with ultradiscretization, a mathematical formalism from tropical geometry providing continuous relaxations of discrete dynamics. The central insight is that the max-plus semiring underlying ultradiscretization naturally models neural threshold dynamics: the log-sum-exp function serves as a differentiable soft-maximum that converges to hard thresholding as a learnable temperature parameter eps to 0. Two neuron models are derived from distinct dynamical systems: UltraLIF from the LIF ordinary differential equation (temporal dynamics) and UltraDLIF from the diffusion equation modeling gap junction coupling across neuronal populations (spatial dynamics). Both yield fully differentiable SNNs trainable via standard backpropagation with no forward-backward mismatch. Theoretical analysis establishes pointwise convergence to classical LIF dynamics with quantitative error bounds and bounded non-vanishing gradients. Experiments on six benchmarks spanning static images, neuromorphic vision, and audio demonstrate improvements over surrogate gradient baselines, with gains most pronounced in single-timestep (T{=}1) settings on neuromorphic and temporal datasets. An optional sparsity penalty enables significant energy reduction while maintaining competitive accuracy.

Verifying the provenance of content is crucial to the function of many organizations, e.g., educational institutions, social media platforms, firms, etc. This problem is becoming increasingly difficult as text generated by Large Language Models (LLMs) becomes almost indistinguishable from human-generated content. In addition, many institutions utilize in-house LLMs and want to ensure that external, non-sanctioned LLMs do not produce content within the institution. In this paper, we answer the following question: Given a piece of text, can we identify whether it was produced by LLM A or B (where B can be a human)? We model LLM-generated text as a sequential stochastic process with complete dependence on history and design zero-shot statistical tests to distinguish between (i) the text generated by two different sets of LLMs A (in-house) and B (non-sanctioned) and also (ii) LLM-generated and human-generated texts. We prove that the type I and type II errors for our tests decrease exponentially in the text length. In designing our tests, we derive concentration inequalities on the difference between log-perplexity and the average entropy of the string under A. Specifically, for a given string, we demonstrate that if the string is generated by A, the log-perplexity of the string under A converges to the average entropy of the string under A, except with an exponentially small probability in string length. We also show that if B generates the text, except with an exponentially small probability in string length, the log-perplexity of the string under A converges to the average cross-entropy of B and A. Lastly, we present preliminary experimental results to support our theoretical results. By enabling guaranteed (with high probability) finding of the origin of harmful LLM-generated text with arbitrary size, we can help combat misinformation.

We study simplified bootstrap problems for probability distributions on the infinite line and the circle. We show that the rapid convergence of the bootstrap method for problems on the infinite line is related to the fact that the smallest eigenvalue of the positive matrices in the exact solution becomes exponentially small for large matrices, while the moments grow factorially. As a result, the positivity condition is very finely tuned. For problems on the circle we show instead that the entries of the positive matrix of Fourier modes of the distribution depend linearly on the initial data of the recursion, with factorially growing coefficients. By positivity, these matrix elements are bounded in absolute value by one, so the initial data must also be fine-tuned. Additionally, we find that we can largely bypass the semi-definite program (SDP) nature of the problem on a circle by recognizing that these Fourier modes must be asymptotically exponentially small. With a simple ansatz, which we call the shoestring bootstrap, we can efficiently identify an interior point of the set of allowed matrices with much higher precision than conventional SDP bounds permit. We apply this method to solving unitary matrix model integrals by numerically constructing the orthogonal polynomials associated with the circle distribution.

Recent studies have suggested frequency-domain Data augmentation (DA) is effec tive for time series prediction. Existing frequency-domain augmentations disturb the original data with various full-spectrum noises, leading to excess domain gap between augmented and original data. Although impressive performance has been achieved in certain cases, frequency-domain DA has yet to be generalized to time series prediction datasets. In this paper, we found that frequency-domain augmentations can be significantly improved by two modifications that limit the perturbations. First, we found that limiting the perturbation to only dominant frequencies significantly outperforms full-spectrum perturbations. Dominant fre quencies represent the main periodicity and trends of the signal and are more important than other frequencies. Second, we found that simply shuffling the dominant frequency components is superior over sophisticated designed random perturbations. Shuffle rearranges the original components (magnitudes and phases) and limits the external noise. With these two modifications, we proposed dominant shuffle, a simple yet effective data augmentation for time series prediction. Our method is very simple yet powerful and can be implemented with just a few lines of code. Extensive experiments with eight datasets and six popular time series models demonstrate that our method consistently improves the baseline performance under various settings and significantly outperforms other DA methods. Code can be accessed at https://kaizhao.net/time-series.

Several machine learning models, including neural networks, consistently misclassify adversarial examples---inputs formed by applying small but intentionally worst-case perturbations to examples from the dataset, such that the perturbed input results in the model outputting an incorrect answer with high confidence. Early attempts at explaining this phenomenon focused on nonlinearity and overfitting. We argue instead that the primary cause of neural networks' vulnerability to adversarial perturbation is their linear nature. This explanation is supported by new quantitative results while giving the first explanation of the most intriguing fact about them: their generalization across architectures and training sets. Moreover, this view yields a simple and fast method of generating adversarial examples. Using this approach to provide examples for adversarial training, we reduce the test set error of a maxout network on the MNIST dataset.

Hardware-aware training (HAT) is widely used to improve the robustness of neural networks on non-ideal AI accelerators, such as analog in-memory computing (IMC) systems. However, not all hardware-induced distortions are equally compensable by training. This paper presents a diagnostic framework that models hardware non-idealities as structured perturbations of the forward operator and evaluates their compatibility with gradient-based optimization. We analyze six representative perturbation classes--read noise, variability, drift, stuck-at faults, IR-drop, and ADC discretization--and identify three key diagnostics: gradient expectation consistency, bounded gradient variance, and non-degenerate sensitivity. Our results show a clear separation between perturbations that can be compensated by HAT and those that consistently break optimization. This provides practical guidance for hardware-software co-design, clarifying which non-idealities can be addressed at the training level and which require circuit-, architecture-, or calibration-level mitigation. This study should be interpreted as a controlled empirical analysis under vanilla forward-perturbation HAT, rather than as a universal theory of hardware-aware training.

We address hallucination detection in Large Language Models (LLMs) and Multimodal Large Language Models (MLLMs) by framing the problem through the lens of dynamical systems stability theory. Rather than treating hallucination as a straightforward classification task, we conceptualize (M)LLMs as dynamical systems, where factual knowledge is represented by stable equilibrium points within the representation space. Our main insight is that hallucinations tend to arise at the boundaries of knowledge-transition regions separating stable and unstable zones. To capture this phenomenon, we propose Lyapunov Probes: lightweight networks trained with derivative-based stability constraints that enforce a monotonic decay in confidence under input perturbations. By performing systematic perturbation analysis and applying a two-stage training process, these probes reliably distinguish between stable factual regions and unstable, hallucination-prone regions. Experiments on diverse datasets and models demonstrate consistent improvements over existing baselines.

Recent advancements in diffusion models revolutionize image generation but pose risks of misuse, such as replicating artworks or generating deepfakes. Existing image protection methods, though effective, struggle to balance protection efficacy, invisibility, and latency, thus limiting practical use. We introduce perturbation pre-training to reduce latency and propose a mixture-of-perturbations approach that dynamically adapts to input images to minimize performance degradation. Our novel training strategy computes protection loss across multiple VAE feature spaces, while adaptive targeted protection at inference enhances robustness and invisibility. Experiments show comparable protection performance with improved invisibility and drastically reduced inference time. The code and demo are available at https://webtoon.github.io/impasto